Geometry. Class Examples (July 29) Paul Yiu. Department of Mathematics Florida Atlantic University. Summer 2014
|
|
- Robert Pierce
- 5 years ago
- Views:
Transcription
1 Geometry lass Examples (July 29) Paul Yiu Department of Mathematics Florida tlantic University c a Summer 2014
2 1 The Pythagorean Theorem Theorem (Pythagoras). The lengths a <cof the sides of a right triangle satisfy the relation a = c 2. Proof. a a a a a 3 1 a Theorem (onverse of Pythagoras theorem). If the lengths of the sides of satisfy a = c 2, then the triangle has a right angle at. Y a c a Z X Proof. onsider a right triangle XY Z with Z =90, YZ = a, and XZ =. y the Pythagorean theorem, XY 2 = YZ 2 +XZ 2 = a = c 2 = 2. It follows that XY =, and XY Z y the SSS test, and = Z =90.
3 2 Theorem (The law of sines). Let R denote the circumradius of a triangle. 2R = a sin α = sin β = c sin γ. α F O E O α D D Since the area of a triangle is given y Δ= 1 2 c sin α, the circumradius can e written as R = ac 4Δ.
4 3 The law of cosines Given a triangle, we denote y a,, c the lengths of the sides,, respectively. Theorem (The law of cosines). c 2 = a a cos γ. c c X a a X Proof. Let X e the altitude on. c 2 = X 2 + X 2 =(a cos γ) 2 +(sin γ) 2 = a 2 2a cos γ + 2 (cos 2 γ +sin 2 γ) = a a cos γ.
5 4 pollonius Theorem Theorem. Given triangle, let D e the midpoint of. The length of the median D is given y =2(D 2 + D 2 ). D Proof. pplying the law of cosines to triangles D and D, and noting that cos D = cos D,wehave 2 = D 2 + D 2 2D D cos D; 2 = D 2 + D 2 2D D cos D, 2 = D 2 + D 2 +2D D cos D. The result follows y adding the first and the third lines. If m a denotes the length of the median on the side, m 2 a = 1 4 (22 +2c 2 a 2 ).
6 ngle isector theorem Theorem (ngle isector theorem). The isectors of an angle of a triangle divide its opposite side in the ratio of the remaining sides. If X and X respectively the internal and external isectors of angle, then X : X = c : and X : X = c :. Z c Z X X Proof. onstruct lines through parallel to the isectors X and X to intersect the line at Z and Z. (1) Note that Z = X = X = Z. This means Z =. learly, X : X = : Z = : = c :. (2) Similarly, Z =, and X : X = : Z = : = c :.
7 6 The lengths of the isectors Theorem. (a) The lengths of the internal and external isectors of angle are respectively t a = 2c + c cos α 2 and t a = 2c c sin α 2. c t a t a X X Remarks. (1) 2c +c is the harmonic mean of and c. It can e constructed as follows. If the perpendicular to X at X intersects and at Y and Z, then Y = Z = 2c +c. c t a Z X Y (2) pplying Stewart s Theorem with λ = c and μ = ±, we also otain the following expressions for the lengths of the angle isectors: ( ( ) ) 2 a t 2 a = c 1, + c ( ( ) 2 t 2 a a = c 1). c
8 7 The incircle The internal angle isectors of a triangle are concurrent at the incenter of the triangle. This is the center of the incircle, the circle tangent to the three sides of the triangle. Let the isectors of angles and intersect at I. onsider the pedals of I on the three sides. Since I is on the isector of angle, IX = IZ. Since I is also on the isector of angle, IX = IY. It follows IX = IY = IZ, and the circle, center I, constructed through X, also passes through Y and Z, and is tangent to the three sides of the triangle. s a s a Y Y Z I Z I s c s X s X s c This is called the incircle of triangle, and I the incenter. Let s e the semiperimeter of triangle. The incircle of triangle touches its sides,, at X, Y, Z such that Y =Z = s a, Z =X = s, X =Y = s c. The inradius of triangle is the radius of its incircle. It is given y r = 2Δ a + + c = Δ s.
9 8 The excircles The internal isector of each angle and the external isectors of the remaining two angles are concurrent at an excenter of the triangle. n excircle can e constructed with this as center, tangent to the lines containing the three sides of the triangle. I I c X Y r a Z r a r a The exradii of a triangle with sides a,, c are given y r a = Δ s a, r = Δ s, r c = Δ s c. I a The areas of the triangles I a, I a, and I a are 1 2 ar a, 1 2 r a, and 1 2 cr a respectively. Since Δ= ΔI a +ΔI a +ΔI a, we have Δ= 1 2 r a( a + + c) =r a (s a), from which r a = Δ s a.
10 9 Heron s formula for the area of a triangle onsider a triangle with area Δ. Denote y r the inradius, and r a the radius of the excircle on the side of triangle. It is convenient to introduce the semiperimeter s = 1 2 (a + + c). I a I r a r Y Y (1) From the similarity of triangles IY and I Y, r = s a. r a s (2) From the similarity of triangles IY and I Y, r r a =(s )(s c). (3) From these, (s a)(s )(s c) r =, s s(s )(s c) r a =. s a Theorem (Heron s formula). Δ= s(s a)(s )(s c). Proof. Δ=rs.
11 10 Menelaus theorem Theorem (Menelaus). Given a triangle with points X, Y, Z on the side lines,, respectively, the points X, Y, Z are collinear if and only if X X Y Y Z Z = 1. Y Z W X Proof. (= ) Let W e the point on such that W//XY. Then, X X = WY Y, and Z Z = Y YW. It follows that X X Y Y Z Z = WY Y Y Y Y YW = Y Y Y Y WY YW = 1. ( =) Suppose the line joining X and Z intersects at Y. From aove, X X Y Y Z = 1 =X Z X Y Y Z Z. It follows that Y Y = Y Y. The points Y and Y divide the segment in the same ratio. These must e the same point, and X, Y, Z are collinear.
12 11 eva s theorem Theorem (eva). Given a triangle with points X, Y, Z on the side lines,, respectively, the lines X, Y, Z are concurrent if and only if X X Y Y Z Z =+1. Z Y P X Proof. (= ) Suppose the lines X, Y, Z intersect at a point P. onsider the line PY cutting the sides of triangle X. y Menelaus theorem, Y Y P PX X = 1, or Y Y P XP X =+1. lso, consider the line PZ cutting the sides of triangle X. y Menelaus theorem again, Z Z X XP P = 1, or Z Z X XP P =+1. Multiplying the two equations together, we have ( =) Exercise. Y Y Z Z X X =+1.
13 1 arycentric coordinates on a line w X v Let and e two fixed points. point X on the line is defined y one of the following: (1) the division ratio X X = w v, (2) distriuting a unit mass at and so as to alance at X: the asolute arycentric coordinates X = v + w v + w, (3) the homogeneous arycentric coordinates relative to : X = v : w.
14 2 arycentric coordinates with respect to a triangle Given a reference triangle, we put at the vertices,, masses u, v, w respectively, and determine the alance point. (u) (u) v + w P (u + v + w) w v u (v) (w) X(v + w) X The masses at and can e replaced y a single mass v + w at the point X = v + w. v + w Together with the mass at, this can e replaced y a mass u + v + w at the point P which divides X in the ratio P : PX = v + w : u. This is the point with asolute arycentric coordinates provided u + v + w 0. u + v + w, u + v + w We also say that the alance point P has homogeneous arycentric coordinates (u : v : w) with reference to.
15 13 evian triangle Theorem (eva). Let X, Y, Z e points on the lines,, respectively. The lines X, Y, Z are collinear if and only if the given points have coordinates of the form X = (0 : y : z), Y = (x : 0 : z), Z = (x : y : 0), for some x, y, z. If this condition is satisfied, the common point of the lines X, Y, Z is P =(x : y : z). Z Y P P P P X P (1) The points X, Y, Z are called the traces of P. We also say that XY Z is the cevian triangle of P (with reference to triangle ). Sometimes, we shall adopt the more functional notation for the cevian triangle and its vertices: cev(p ): P =(0:y : z), P =(x :0:z), P =(x : y :0). (2) The point P divides the segment X in the ratio PX : X = x : x + y + z.
16 8 Theorem. The homogeneous arycentric coordinates of P are in the proportion of the (signed) areas of triangles P, P, P. P P P P P X Proof. Since PX : X = x : x + y + z, ΔPX : ΔX = x : x + y + z, and ΔPX : ΔX = x : x + y + z. Therefore, Similarly, ΔP = From these, ΔP = ΔPX +ΔPX x = x + y + z ΔX + x x + y + z ΔX x = (ΔX +ΔX) x + y + z x = x + y + z Δ. y x + y + z Δ, ΔP = z x + y + z Δ. ΔP : ΔP : ΔP = x : y : z.
17 10 rea and arycentric coordinates Theorem. If for i =1, 2, 3, P i = x i +y i +z i (in asolute arycentric coordinates), then the area of the oriented triangle P 1 P 2 P 3 is x 1 y 1 z 1 ΔP 1 P 2 P 3 = x 2 y 2 z 2 x 3 y 3 z 3 Δ.
18 9 Equations of straight lines Two-point form The area formula has an easy and extremely important consequence: the equation of the line joining two points with coordinates (x 1 : y 1 : z 1 ) and (x 2 : y 2 : z 2 ) is x 1 y 1 z 1 x 2 y 2 z 2 x y z =0, or (y 1 z 2 y 2 z 1 )x +(z 1 x 2 z 2 x 1 )y +(x 1 y 2 x 2 y 1 )z =0. Examples (1) The equations of the sidelines,, are respectively x =0, y =0, z =0. (2) Given a point P =(u : v : w), the cevian line P has equation wy vz =0; similarly for the other two cevian lines P and P. These lines intersect corresponding sidelines at the traces of P : P =(0:v : w), P =(u :0:w), P =(u : v :0). (3) The equation of the line joining the centroid and the incenter is a c x y z =0, or ( c)x +(c a)y +(a )z =0.
19 10 Intersection of two lines The intersection of the two lines p 1 x + q 1 y + r 1 z =0, p 2 x + q 2 y + r 2 z =0 is the point (q 1 r 2 q 2 r 1 : r 1 p 2 r 2 p 1 : p 1 q 2 p 2 q 1 ). Proposition 0.1. Three lines p i x + q i y + r i z =0, i =1, 2, 3, are concurrent if and only if p 1 q 1 r 1 p 2 q 2 r 2 p 3 q 3 r 3 =0.
20 11 The infinite point of a line The infinite point of a line L has homogeneous coordinates given y the difference of the asolute arycentric coordinates of two distinct points on the line. s such, the coordinate sum of an infinite point is zero. We think of all infinite points constituting the line at infinity, L, which has equation x + y + z =0. Examples 1. The infinite points of the side lines,, are (0 : 1 :1), (1 : 0 : 1), ( 1 :1:0)respectively. 2. The infinite point of the altitude has homogeneous coordinates (0 : S : S ) a 2 (1:0:0)=( a 2 : S : S ). 3. More generally, the infinite point of the line px + qy + rz =0is (q r : r p : p q).
21 12 Parallel lines Parallel lines have the same infinite point. The line through P =(u : v : w) parallel to L : px + qy + rz =0has equation q r r p p q u v w x y z =0. 1. Find the equations of the lines through P =(u : v : w) parallel to the side lines. 2. Let DEF e the medial triangle of, and P a point with cevian triangle XY Z (with respect to. Find P such that the lines DX, EY, FZ are parallel to the internal isectors of angles,, respectively. 1 1 The Nagel point P =( + c a : c + a : a + c).
22 15 Solution to Example (5, 8, 9) triangle. is a triangle with a =5, =8, and c =9. Show that the Gergonne point lies on the line joining the midpoints of the sides and. Y X 5 D G e I E 8 Solution. s =11. Y = Z = s c =2, Y = s a =6, Z = s =3. Z 9 solute arycentric coordinates: D = + 2, E = + 2. Homogeneous arycentric coordinates: X = 0 : 2 : 3 Y = 1 : 0 : 3 G e = 1 : 2 : 3 Therefore G e = The Gergonne point lies on the line DE if and only if G e = t D +(1 t)e for some t: = t + 2 This is the case with t = (1 t) + 2 = 1 t 2 + t
23 16 Example (7, 8, 9) triangle. is a triangle with a =7, =8, and c =9. Show that the Nagel point lies on the line joining the points of tangency of incircle with the sides and. 9 Z Z N a Y Y 8 7
24 17 Solution to Example (7, 8, 9) triangle. is a triangle with a =7, =8, and c =9. Show that the Nagel point lies on the line joining the points of tangency of incircle with the sides and. 9 Z Z N a Y Y 8 Solution. s =12. Z = s =4, Y = s c =3; Z =4, Y =3. solute arycentric coordinates: Y = 3+5 8, Z = Homogeneous arycentric coordinates: 7 Y = 5 : 0 : 3 Z = 5 : 4 : 0 N a = 5 : 4 : 3 Therefore N a = The Nagel point lies on the line YZif and only if N a = t Y +(1 t)z for some t: = t 3 +(1 t) 4 = This is the case with t = t 5(1 t) + + 5t
25
Geometry. Class Examples (July 8) Paul Yiu. Department of Mathematics Florida Atlantic University. Summer 2014
Geometry lass Examples (July 8) Paul Yiu Department of Mathematics Florida tlantic University c b a Summer 2014 1 The incircle The internal angle bisectors of a triangle are concurrent at the incenter
More informationChapter 6. Basic triangle centers. 6.1 The Euler line The centroid
hapter 6 asic triangle centers 6.1 The Euler line 6.1.1 The centroid Let E and F be the midpoints of and respectively, and G the intersection of the medians E and F. onstruct the parallel through to E,
More informationGeometry. Class Examples (July 10) Paul Yiu. Department of Mathematics Florida Atlantic University. Summer 2014
Geometry lass Examples (July 10) Paul iu Department of Mathematics Florida tlantic University c b a Summer 2014 1 Menelaus theorem Theorem (Menelaus). Given a triangle with points,, on the side lines,,
More informationChapter 5. Menelaus theorem. 5.1 Menelaus theorem
hapter 5 Menelaus theorem 5.1 Menelaus theorem Theorem 5.1 (Menelaus). Given a triangle with points,, on the side lines,, respectively, the points,, are collinear if and only if = 1. W Proof. (= ) LetW
More informationMenelaus and Ceva theorems
hapter 21 Menelaus and eva theorems 21.1 Menelaus theorem Theorem 21.1 (Menelaus). Given a triangle with points,, on the side lines,, respectively, the points,, are collinear if and only if = 1. W Proof.
More informationGeometry. Class Examples (July 3) Paul Yiu. Department of Mathematics Florida Atlantic University. Summer 2014
Geometry lass Examples (July 3) Paul Yiu Department of Mathematics Florida tlantic University c b a Summer 2014 Example 11(a): Fermat point. Given triangle, construct externally similar isosceles triangles
More informationAdvanced Euclidean Geometry
dvanced Euclidean Geometry Paul iu Department of Mathematics Florida tlantic University Summer 2016 July 11 Menelaus and eva Theorems Menelaus theorem Theorem 0.1 (Menelaus). Given a triangle with points,,
More informationThe circumcircle and the incircle
hapter 4 The circumcircle and the incircle 4.1 The Euler line 4.1.1 nferior and superior triangles G F E G D The inferior triangle of is the triangle DEF whose vertices are the midpoints of the sides,,.
More informationHomogeneous Barycentric Coordinates
hapter 9 Homogeneous arycentric oordinates 9. bsolute and homogeneous barycentric coordinates The notion of barycentric coordinates dates back to. F. Möbius ( ). Given a reference triangle, we put at the
More informationConstruction of a Triangle from the Feet of Its Angle Bisectors
onstruction of a Triangle from the Feet of Its ngle isectors Paul Yiu bstract. We study the problem of construction of a triangle from the feet of its internal angle bisectors. conic solution is possible.
More informationSurvey of Geometry. Paul Yiu. Department of Mathematics Florida Atlantic University. Spring 2007
Survey of Geometry Paul Yiu Department of Mathematics Florida tlantic University Spring 2007 ontents 1 The circumcircle and the incircle 1 1.1 The law of cosines and its applications.............. 1 1.2
More informationGeometry. Class Examples (July 1) Paul Yiu. Department of Mathematics Florida Atlantic University. Summer 2014
Geometry lass Examples (July 1) Paul Yiu Department of Mathematics Florida tlantic University c b a Summer 2014 21 Example 11: Three congruent circles in a circle. The three small circles are congruent.
More informationMenelaus and Ceva theorems
hapter 3 Menelaus and eva theorems 3.1 Menelaus theorem Theorem 3.1 (Menelaus). Given a triangle with points,, on the side lines,, respectively, the points,, are collinear if and only if = 1. W Proof.
More informationThe Menelaus and Ceva Theorems
hapter 7 The Menelaus and eva Theorems 7.1 7.1.1 Sign convention Let and be two distinct points. point on the line is said to divide the segment in the ratio :, positive if is between and, and negative
More informationConic Construction of a Triangle from the Feet of Its Angle Bisectors
onic onstruction of a Triangle from the Feet of Its ngle isectors Paul Yiu bstract. We study an extension of the problem of construction of a triangle from the feet of its internal angle bisectors. Given
More informationGeometry. Class Examples (July 1) Paul Yiu. Department of Mathematics Florida Atlantic University. Summer 2014
Geometry lass Examples (July 1) Paul Yiu Department of Mathematics Florida tlantic University c b a Summer 2014 1 Example 1(a). Given a triangle, the intersection P of the perpendicular bisector of and
More informationIsotomic Inscribed Triangles and Their Residuals
Forum Geometricorum Volume 3 (2003) 125 134. FORUM GEOM ISSN 1534-1178 Isotomic Inscribed Triangles and Their Residuals Mario Dalcín bstract. We prove some interesting results on inscribed triangles which
More informationA Note on Reflections
Forum Geometricorum Volume 14 (2014) 155 161. FORUM GEOM SSN 1534-1178 Note on Reflections Emmanuel ntonio José García bstract. We prove some simple results associated with the triangle formed by the reflections
More informationSurvey of Geometry. Supplementary Notes on Elementary Geometry. Paul Yiu. Department of Mathematics Florida Atlantic University.
Survey of Geometry Supplementary Notes on Elementary Geometry Paul Yiu Department of Mathematics Florida tlantic University Summer 2007 ontents 1 The Pythagorean theorem i 1.1 The hypotenuse of a right
More informationChapter 3. The angle bisectors. 3.1 The angle bisector theorem
hapter 3 The angle bisectors 3.1 The angle bisector theorem Theorem 3.1 (ngle bisector theorem). The bisectors of an angle of a triangle divide its opposite side in the ratio of the remaining sides. If
More informationXIII GEOMETRICAL OLYMPIAD IN HONOUR OF I.F.SHARYGIN The correspondence round. Solutions
XIII GEOMETRIL OLYMPID IN HONOUR OF I.F.SHRYGIN The correspondence round. Solutions 1. (.Zaslavsky) (8) Mark on a cellular paper four nodes forming a convex quadrilateral with the sidelengths equal to
More informationChapter 8. Feuerbach s theorem. 8.1 Distance between the circumcenter and orthocenter
hapter 8 Feuerbach s theorem 8.1 Distance between the circumcenter and orthocenter Y F E Z H N X D Proposition 8.1. H = R 1 8 cosαcos β cosγ). Proof. n triangle H, = R, H = R cosα, and H = β γ. y the law
More informationChapter 2. The laws of sines and cosines. 2.1 The law of sines. Theorem 2.1 (The law of sines). Let R denote the circumradius of a triangle ABC.
hapter 2 The laws of sines and cosines 2.1 The law of sines Theorem 2.1 (The law of sines). Let R denote the circumradius of a triangle. 2R = a sin α = b sin β = c sin γ. α O O α as Since the area of a
More informationThe Apollonius Circle and Related Triangle Centers
Forum Geometricorum Qolume 3 (2003) 187 195. FORUM GEOM ISSN 1534-1178 The Apollonius Circle and Related Triangle Centers Milorad R. Stevanović Abstract. We give a simple construction of the Apollonius
More informationConstruction of Ajima Circles via Centers of Similitude
Forum Geometricorum Volume 15 (2015) 203 209. FORU GEO SS 1534-1178 onstruction of jima ircles via enters of Similitude ikolaos Dergiades bstract. We use the notion of the centers of similitude of two
More informationSOME NEW THEOREMS IN PLANE GEOMETRY. In this article we will represent some ideas and a lot of new theorems in plane geometry.
SOME NEW THEOREMS IN PLNE GEOMETRY LEXNDER SKUTIN 1. Introduction arxiv:1704.04923v3 [math.mg] 30 May 2017 In this article we will represent some ideas and a lot of new theorems in plane geometry. 2. Deformation
More informationOn Emelyanov s Circle Theorem
Journal for Geometry and Graphics Volume 9 005, No., 55 67. On Emelyanov s ircle Theorem Paul Yiu Department of Mathematical Sciences, Florida Atlantic University Boca Raton, Florida, 3343, USA email:
More informationCalgary Math Circles: Triangles, Concurrency and Quadrilaterals 1
Calgary Math Circles: Triangles, Concurrency and Quadrilaterals 1 1 Triangles: Basics This section will cover all the basic properties you need to know about triangles and the important points of a triangle.
More informationXIV GEOMETRICAL OLYMPIAD IN HONOUR OF I.F.SHARYGIN The correspondence round. Solutions
XIV GEOMETRIL OLYMPI IN HONOUR OF I.F.SHRYGIN The correspondence round. Solutions 1. (L.Shteingarts, grade 8) Three circles lie inside a square. Each of them touches externally two remaining circles. lso
More informationRecreational Mathematics
Recreational Mathematics Paul Yiu Department of Mathematics Florida Atlantic University Summer 2003 Chapters 5 8 Version 030630 Chapter 5 Greatest common divisor 1 gcd(a, b) as an integer combination of
More informationOn the Circumcenters of Cevasix Configurations
Forum Geometricorum Volume 3 (2003) 57 63. FORUM GEOM ISSN 1534-1178 On the ircumcenters of evasix onfigurations lexei Myakishev and Peter Y. Woo bstract. We strengthen Floor van Lamoen s theorem that
More informationHeptagonal Triangles and Their Companions
Forum Geometricorum Volume 9 (009) 15 148. FRUM GEM ISSN 1534-1178 Heptagonal Triangles and Their ompanions Paul Yiu bstract. heptagonal triangle is a non-isosceles triangle formed by three vertices of
More informationForum Geometricorum Volume 13 (2013) 1 6. FORUM GEOM ISSN Soddyian Triangles. Frank M. Jackson
Forum Geometricorum Volume 3 (203) 6. FORUM GEOM ISSN 534-78 Soddyian Triangles Frank M. Jackson bstract. Soddyian triangle is a triangle whose outer Soddy circle has degenerated into a straight line.
More informationSingapore International Mathematical Olympiad Training Problems
Singapore International athematical Olympiad Training Problems 18 January 2003 1 Let be a point on the segment Squares D and EF are erected on the same side of with F lying on The circumcircles of D and
More informationPlane geometry Circles: Problems with some Solutions
The University of Western ustralia SHL F MTHMTIS & STTISTIS UW MY FR YUNG MTHMTIINS Plane geometry ircles: Problems with some Solutions 1. Prove that for any triangle, the perpendicular bisectors of the
More informationSteiner s porism and Feuerbach s theorem
hapter 10 Steiner s porism and Feuerbach s theorem 10.1 Euler s formula Lemma 10.1. f the bisector of angle intersects the circumcircle at M, then M is the center of the circle through,,, and a. M a Proof.
More informationChapter 1. Theorems of Ceva and Menelaus
hapter 1 Theorems of eva and Menelaus We start these lectures by proving some of the most basic theorems in the geometry of a planar triangle. Let,, be the vertices of the triangle and,, be any points
More informationA Note on the Barycentric Square Roots of Kiepert Perspectors
Forum Geometricorum Volume 6 (2006) 263 268. FORUM GEOM ISSN 1534-1178 Note on the arycentric Square Roots of Kiepert erspectors Khoa Lu Nguyen bstract. Let be an interior point of a given triangle. We
More informationGeneralized Archimedean Arbelos Twins
Forum Geometricorum Volume 4 (204) 409 48 FORUM GEOM ISSN 534-78 Generalized rchimedean rbelos Twins Nikolaos Dergiades bstract We generalize the well known rchimedean arbelos twins by extending the notion
More informationHarmonic Division and its Applications
Harmonic ivision and its pplications osmin Pohoata Let d be a line and,,, and four points which lie in this order on it. he four-point () is called a harmonic division, or simply harmonic, if =. If is
More informationGeometry JWR. Monday September 29, 2003
Geometry JWR Monday September 29, 2003 1 Foundations In this section we see how to view geometry as algebra. The ideas here should be familiar to the reader who has learned some analytic geometry (including
More informationXI Geometrical Olympiad in honour of I.F.Sharygin Final round. Grade 8. First day. Solutions Ratmino, 2015, July 30.
XI Geometrical Olympiad in honour of I.F.Sharygin Final round. Grade 8. First day. Solutions Ratmino, 2015, July 30. 1. (V. Yasinsky) In trapezoid D angles and are right, = D, D = + D, < D. Prove that
More informationA Distance Property of the Feuerbach Point and Its Extension
Forum Geometricorum Volume 16 (016) 83 90. FOUM GEOM ISSN 1534-1178 A Distance Property of the Feuerbach Point and Its Extension Sándor Nagydobai Kiss Abstract. We prove that among the distances from the
More informationXII Geometrical Olympiad in honour of I.F.Sharygin Final round. Solutions. First day. 8 grade
XII Geometrical Olympiad in honour of I.F.Sharygin Final round. Solutions. First day. 8 grade Ratmino, 2016, July 31 1. (Yu.linkov) n altitude H of triangle bisects a median M. Prove that the medians of
More informationBarycentric coordinates
arycentric coordinates by Paris Pamfilos The artist finds a greater pleasure in painting than in having completed the picture. Seneca, Letter to Lucilius ontents 1 Preliminaries and definition 2 2 Traces,
More informationClassical Theorems in Plane Geometry 1
BERKELEY MATH CIRCLE 1999 2000 Classical Theorems in Plane Geometry 1 Zvezdelina Stankova-Frenkel UC Berkeley and Mills College Note: All objects in this handout are planar - i.e. they lie in the usual
More informationProblem Solving and Recreational Mathematics
Problem Solving and Recreational Mathematics Paul Yiu Department of Mathematics Florida tlantic University Summer 2012 Chapters 16 33 July 20 Monday 6/25 7/2 7/9 7/16 7/23 7/30 Wednesday 6/27 *** 7/11
More informationTheorem 1.2 (Converse of Pythagoras theorem). If the lengths of the sides of ABC satisfy a 2 + b 2 = c 2, then the triangle has a right angle at C.
hapter 1 Some asic Theorems 1.1 The ythagorean Theorem Theorem 1.1 (ythagoras). The lengths a b < c of the sides of a right triangle satisfy the relation a + b = c. roof. b a a 3 b b 4 b a b 4 1 a a 3
More informationIntroduction to Number Theory
Introduction to Number Theory Paul Yiu Department of Mathematics Florida Atlantic University Spring 017 March 7, 017 Contents 10 Pythagorean and Heron triangles 57 10.1 Construction of Pythagorean triangles....................
More informationIsogonal Conjugates. Navneel Singhal October 9, Abstract
Isogonal Conjugates Navneel Singhal navneel.singhal@ymail.com October 9, 2016 Abstract This is a short note on isogonality, intended to exhibit the uses of isogonality in mathematical olympiads. Contents
More informationDistances Among the Feuerbach Points
Forum Geometricorum Volume 16 016) 373 379 FORUM GEOM ISSN 153-1178 Distances mong the Feuerbach Points Sándor Nagydobai Kiss bstract We find simple formulas for the distances from the Feuerbach points
More informationThree Natural Homoteties of The Nine-Point Circle
Forum Geometricorum Volume 13 (2013) 209 218. FRUM GEM ISS 1534-1178 Three atural omoteties of The ine-point ircle Mehmet Efe kengin, Zeyd Yusuf Köroğlu, and Yiğit Yargiç bstract. Given a triangle with
More informationTriangles III. Stewart s Theorem (1746) Stewart s Theorem (1746) 9/26/2011. Stewart s Theorem, Orthocenter, Euler Line
Triangles III Stewart s Theorem, Orthocenter, uler Line 23-Sept-2011 M 341 001 1 Stewart s Theorem (1746) With the measurements given in the triangle below, the following relationship holds: a 2 n + b
More informationBicevian Tucker Circles
Forum Geometricorum Volume 7 (2007) 87 97. FORUM GEOM ISSN 1534-1178 icevian Tucker ircles ernard Gibert bstract. We prove that there are exactly ten bicevian Tucker circles and show several curves containing
More informationXIV Geometrical Olympiad in honour of I.F.Sharygin Final round. Solutions. First day. 8 grade
XIV Geometrical Olympiad in honour of I.F.Sharygin Final round. Solutions. First day. 8 grade 1. (M.Volchkevich) The incircle of right-angled triangle A ( = 90 ) touches at point K. Prove that the chord
More informationThe Apollonius Circle as a Tucker Circle
Forum Geometricorum Volume 2 (2002) 175 182 FORUM GEOM ISSN 1534-1178 The Apollonius Circle as a Tucker Circle Darij Grinberg and Paul Yiu Abstract We give a simple construction of the circular hull of
More informationSMT 2018 Geometry Test Solutions February 17, 2018
SMT 018 Geometry Test Solutions February 17, 018 1. Consider a semi-circle with diameter AB. Let points C and D be on diameter AB such that CD forms the base of a square inscribed in the semicircle. Given
More informationTrigonometric Fundamentals
1 Trigonometric Fundamentals efinitions of Trigonometric Functions in Terms of Right Triangles Let S and T be two sets. function (or mapping or map) f from S to T (written as f : S T ) assigns to each
More informationPower Round: Geometry Revisited
Power Round: Geometry Revisited Stobaeus (one of Euclid s students): But what shall I get by learning these things? Euclid to his slave: Give him three pence, since he must make gain out of what he learns.
More informationMAHESH TUTORIALS. Time : 1 hr. 15 min. Q.1. Solve the following : 3
S.S.. MHESH TUTRILS Test - II atch : S Marks : 30 Date : GEMETRY hapter : 1,, 3 Time : 1 hr. 15 min..1. Solve the following : 3 The areas of two similar triangles are 18 cm and 3 cm respectively. What
More informationCircle Chains Inside a Circular Segment
Forum eometricorum Volume 9 (009) 73 79. FRUM EM ISSN 534-78 ircle hains Inside a ircular Segment iovanni Lucca bstract. We consider a generic circles chain that can be drawn inside a circular segment
More informationGenealogy of Pythagorean triangles
Chapter 0 Genealogy of Pythagorean triangles 0. Two ternary trees of rational numbers Consider the rational numbers in the open interval (0, ). Each of these is uniquely in the form q, for relatively prime
More information22 SAMPLE PROBLEMS WITH SOLUTIONS FROM 555 GEOMETRY PROBLEMS
22 SPL PROLS WITH SOLUTIOS FRO 555 GOTRY PROLS SOLUTIOS S O GOTRY I FIGURS Y. V. KOPY Stanislav hobanov Stanislav imitrov Lyuben Lichev 1 Problem 3.9. Let be a quadrilateral. Let J and I be the midpoints
More informationNotes on Barycentric Homogeneous Coordinates
Notes on arycentric Homogeneous oordinates Wong Yan Loi ontents 1 arycentric Homogeneous oordinates 2 2 Lines 3 3 rea 5 4 Distances 5 5 ircles I 8 6 ircles II 10 7 Worked Examples 14 8 Exercises 20 9 Hints
More informationChapter 1. Some Basic Theorems. 1.1 The Pythagorean Theorem
hapter 1 Some asic Theorems 1.1 The ythagorean Theorem Theorem 1.1 (ythagoras). The lengths a b < c of the sides of a right triangle satisfy the relation a 2 + b 2 = c 2. roof. b a a 3 2 b 2 b 4 b a b
More informationChapter 3. Coaxial circles. 3.1 The radical axis of two circles. A quadratic equation of the form
Chapter 3 Coaxial circles 3.1 The radical axis of two circles A quadratic equation of the form x 2 +y 2 +2gx+2fy +c = 0 represents a circle, center( g, f) and radius the square root ofg 2 +f 2 c. It is
More informationMidcircles and the Arbelos
Forum Geometricorum Volume 7 (2007) 53 65. FORUM GEOM ISSN 1534-1178 Midcircles and the rbelos Eric Danneels and Floor van Lamoen bstract. We begin with a study of inversions mapping one given circle into
More informationInequalities for Triangles and Pointwise Characterizations
Inequalities for Triangles and Pointwise haracterizations Theorem (The Scalene Inequality): If one side of a triangle has greater length than another side, then the angle opposite the longer side has the
More informationCollinearity of the First Trisection Points of Cevian Segments
Forum eometricorum Volume 11 (2011) 217 221. FORUM EOM ISSN 154-1178 ollinearity of the First Trisection oints of evian Segments Francisco Javier arcía apitán bstract. We show that the first trisection
More informationOn the Feuerbach Triangle
Forum Geometricorum Volume 17 2017 289 300. FORUM GEOM ISSN 1534-1178 On the Feuerbach Triangle Dasari Naga Vijay Krishna bstract. We study the relations among the Feuerbach points of a triangle and the
More informationSome Collinearities in the Heptagonal Triangle
Forum Geometricorum Volume 16 (2016) 249 256. FRUM GEM ISSN 1534-1178 Some ollinearities in the Heptagonal Triangle bdilkadir ltintaş bstract. With the methods of barycentric coordinates, we establish
More informationTrigonometrical identities and inequalities
Trigonometrical identities and inequalities Finbarr Holland January 1, 010 1 A review of the trigonometrical functions These are sin, cos, & tan. These are discussed in the Maynooth Olympiad Manual, which
More informationarxiv: v1 [math.ho] 10 Feb 2018
RETIVE GEOMETRY LEXNDER SKUTIN arxiv:1802.03543v1 [math.ho] 10 Feb 2018 1. Introduction This work is a continuation of [1]. s in the previous article, here we will describe some interesting ideas and a
More informationChapter 14. Cevian nest theorem Trilinear pole and polar Trilinear polar of a point
hapter 14 evian nest theorem 14.1 Trilinear pole and polar 14.1.1 Trilinear polar of a point Given a point ith traces and on the sidelines of triangle let = Y = Z =. These points Y Z lie on a line called
More informationCollinearity/Concurrence
Collinearity/Concurrence Ray Li (rayyli@stanford.edu) June 29, 2017 1 Introduction/Facts you should know 1. (Cevian Triangle) Let ABC be a triangle and P be a point. Let lines AP, BP, CP meet lines BC,
More informationQ.1 If a, b, c are distinct positive real in H.P., then the value of the expression, (A) 1 (B) 2 (C) 3 (D) 4. (A) 2 (B) 5/2 (C) 3 (D) none of these
Q. If a, b, c are distinct positive real in H.P., then the value of the expression, b a b c + is equal to b a b c () (C) (D) 4 Q. In a triangle BC, (b + c) = a bc where is the circumradius of the triangle.
More informationCLASS IX GEOMETRY MOCK TEST PAPER
Total time:3hrs darsha vidyalay hunashyal P. M.M=80 STION- 10 1=10 1) Name the point in a triangle that touches all sides of given triangle. Write its symbol of representation. 2) Where is thocenter of
More informationThe Lemoine Cubic and Its Generalizations
Forum Geometricorum Volume 2 (2002) 47 63. FRUM GEM ISSN 1534-1178 The Lemoine ubic and Its Generalizations ernard Gibert bstract. For a given triangle, the Lemoine cubic is the locus of points whose cevian
More informationHagge circles revisited
agge circles revisited Nguyen Van Linh 24/12/2011 bstract In 1907, Karl agge wrote an article on the construction of circles that always pass through the orthocenter of a given triangle. The purpose of
More informationDefinitions. (V.1). A magnitude is a part of a magnitude, the less of the greater, when it measures
hapter 8 Euclid s Elements ooks V 8.1 V.1-3 efinitions. (V.1). magnitude is a part of a magnitude, the less of the greater, when it measures the greater. (V.2). The greater is a multiple of the less when
More informationCircles. Exercise 9.1
9 uestion. Exercise 9. How many tangents can a circle have? Solution For every point of a circle, we can draw a tangent. Therefore, infinite tangents can be drawn. uestion. Fill in the blanks. (i) tangent
More informationA quick introduction to (Ceva s and) Menelaus s Theorem
quick introduction to (eva s and) Menelaus s Theorem 1 Introduction Michael Tang May 17, 2015 Menelaus s Theorem, often partnered with eva s Theorem, is a geometric result that determines when three points
More informationTwo applications of the theorem of Carnot
Two applications of the theorem of Carnot Zoltán Szilasi Abstract Using the theorem of Carnot we give elementary proofs of two statements of C Bradley We prove his conjecture concerning the tangents to
More informationHigher Geometry Problems
Higher Geometry Problems (1) Look up Eucidean Geometry on Wikipedia, and write down the English translation given of each of the first four postulates of Euclid. Rewrite each postulate as a clear statement
More information1. The unit vector perpendicular to both the lines. Ans:, (2)
1. The unit vector perpendicular to both the lines x 1 y 2 z 1 x 2 y 2 z 3 and 3 1 2 1 2 3 i 7j 7k i 7j 5k 99 5 3 1) 2) i 7j 5k 7i 7j k 3) 4) 5 3 99 i 7j 5k Ans:, (2) 5 3 is Solution: Consider i j k a
More informationChapter 6. Worked-Out Solutions. Chapter 6 Maintaining Mathematical Proficiency (p. 299)
hapter 6 hapter 6 Maintaining Mathematical Proficiency (p. 99) 1. Slope perpendicular to y = 1 x 5 is. y = x + b 1 = + b 1 = 9 + b 10 = b n equation of the line is y = x + 10.. Slope perpendicular to y
More informationA MOST BASIC TRIAD OF PARABOLAS ASSOCIATED WITH A TRIANGLE
Global Journal of Advanced Research on Classical and Modern Geometries ISSN: 2284-5569, Vol.6, (207), Issue, pp.45-57 A MOST BASIC TRIAD OF PARABOLAS ASSOCIATED WITH A TRIANGLE PAUL YIU AND XIAO-DONG ZHANG
More informationThe Napoleon Configuration
Forum eometricorum Volume 2 (2002) 39 46. FORUM EOM ISSN 1534-1178 The Napoleon onfiguration illes outte bstract. It is an elementary fact in triangle geometry that the two Napoleon triangles are equilateral
More informationDepartment of Mathematical and Statistical Sciences University of Alberta
MATH 214 (R1) Winter 2008 Intermediate Calculus I Solutions to Problem Set #8 Completion Date: Friday March 14, 2008 Department of Mathematical and Statistical Sciences University of Alberta Question 1.
More informationAffine Transformations
Solutions to hapter Problems 435 Then, using α + β + γ = 360, we obtain: ( ) x a = (/2) bc sin α a + ac sin β b + ab sin γ c a ( ) = (/2) bc sin α a 2 + (ac sin β)(ab cos γ ) + (ab sin γ )(ac cos β) =
More informationNagel, Speiker, Napoleon, Torricelli. Centroid. Circumcenter 10/6/2011. MA 341 Topics in Geometry Lecture 17
Nagel, Speiker, Napoleon, Torricelli MA 341 Topics in Geometry Lecture 17 Centroid The point of concurrency of the three medians. 07-Oct-2011 MA 341 2 Circumcenter Point of concurrency of the three perpendicular
More informationHigher Geometry Problems
Higher Geometry Problems (1 Look up Eucidean Geometry on Wikipedia, and write down the English translation given of each of the first four postulates of Euclid. Rewrite each postulate as a clear statement
More informationON THE GERGONNE AND NAGEL POINTS FOR A HYPERBOLIC TRIANGLE
INTERNATIONAL JOURNAL OF GEOMETRY Vol 6 07 No - ON THE GERGONNE AND NAGEL POINTS FOR A HYPERBOLIC TRIANGLE PAUL ABLAGA Abstract In this note we prove the existence of the analogous points of the Gergonne
More information10. Show that the conclusion of the. 11. Prove the above Theorem. [Th 6.4.7, p 148] 4. Prove the above Theorem. [Th 6.5.3, p152]
foot of the altitude of ABM from M and let A M 1 B. Prove that then MA > MB if and only if M 1 A > M 1 B. 8. If M is the midpoint of BC then AM is called a median of ABC. Consider ABC such that AB < AC.
More informationSimple Proofs of Feuerbach s Theorem and Emelyanov s Theorem
Forum Geometricorum Volume 18 (2018) 353 359. FORUM GEOM ISSN 1534-1178 Simple Proofs of Feuerbach s Theorem and Emelyanov s Theorem Nikolaos Dergiades and Tran Quang Hung Abstract. We give simple proofs
More informationTriangle Centers. Maria Nogin. (based on joint work with Larry Cusick)
Triangle enters Maria Nogin (based on joint work with Larry usick) Undergraduate Mathematics Seminar alifornia State University, Fresno September 1, 2017 Outline Triangle enters Well-known centers enter
More informationRevised Edition: 2016 ISBN All rights reserved.
Revised Edition: 2016 ISBN 978-1-280-29557-7 All rights reserved. Published by: Library Press 48 West 48 Street, Suite 1116, New York, NY 10036, United States Email: info@wtbooks.com Table of Contents
More informationTwo applications of the theorem of Carnot
Annales Mathematicae et Informaticae 40 (2012) pp. 135 144 http://ami.ektf.hu Two applications of the theorem of Carnot Zoltán Szilasi Institute of Mathematics, MTA-DE Research Group Equations, Functions
More informationExamples: Identify three pairs of parallel segments in the diagram. 1. AB 2. BC 3. AC. Write an equation to model this theorem based on the figure.
5.1: Midsegments of Triangles NOTE: Midsegments are also to the third side in the triangle. Example: Identify the 3 midsegments in the diagram. Examples: Identify three pairs of parallel segments in the
More informationGeneralized Mandart Conics
Forum Geometricorum Volume 4 (2004) 177 198. FORUM GEOM ISSN 1534-1178 Generalized Mandart onics ernard Gibert bstract. We consider interesting conics associated with the configuration of three points
More information