SOME NEW THEOREMS IN PLANE GEOMETRY. In this article we will represent some ideas and a lot of new theorems in plane geometry.

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1 SOME NEW THEOREMS IN PLNE GEOMETRY LEXNDER SKUTIN 1. Introduction arxiv: v3 [math.mg] 30 May 2017 In this article we will represent some ideas and a lot of new theorems in plane geometry. 2. Deformation of equilateral triangle 2.1. Deformation principle for equilateral triangle. If some triangle points lie on a circle (line) or are equivalent in the case of an equilateral triangle, then in the general case of an arbitrary triangle they are connected by some natural relations. Thus, triangle geometry can be seen as a deformation of the equilateral triangle geometry. This principle partially describes why there exists so many relations between the Kimberling centers X i (see definition of the Kimberling center here [2]) First example of application. onsider an equilateral triangle with center O and let P be an arbitrary point on the circle (). Then we can imagine that O is the first Fermat point of the triangle, and the point P is the second Fermat point of the triangle. onsider a circle ω with center at P and passing through the point O. Let X, Y be the intersection points of the circles () and ω. Then it is easy to see that the line XY passes through the middle of the segment OP. Thus, we can guess that this fact takes place in general for an arbitrary triangle, and it s first and the second Fermat points F 1, F 2. Thus, we obtain the following theorem. Theorem 2.1. onsider triangle with the first Fermat point F 1 and the second Fermat point F 2. onsider a circle ω with center at F 2 and radius F 2 F 1. Then the radical line of circles ω, () goes through the midpoint of the segment F 1 F 2. P F 2 O F Mathematics Subject lassification. 51M04, 51N20. Key words and phrases. Plane geometry, nalytic geometry. 1

2 2 LEXNDER SKUTIN 2.3. Fermat points. Theorem 2.2. onsider triangle with the first Fermat point F 1 and the second Fermat point F 2. onsider a point on, such that F 2 is parallel to F 1. Similarly define,. Then,, lie on the same line which goes through the middle of F 1 F 2. Theorem 2.3. Given triangle with the first Fermat point F 1 and the second Fermat point F 2. Let F be the second Fermat point of F 1, similarly define F, F. (1) Reflect F wrt line F 1 and get the point F, similarly define the points F, F. Then F 2 lies on the circles (F F F ), (F F F ). (2) onsider isogonal conjugations F, F, F of F wrt F 1, F 1, F 1 respectively. Then F F F = π/3. (3) Triangles and F F F are perspective. Theorem 2.4. Let F 1, F 2 be the first and the second Fermat points for. onsider an alphabet with the letters a,b,c. onsider the set W of all words in this alphabet, and let the function denote the length of a word in W. Inductively define F (ω), F(ω), F(ω) : (i) Let F ( ) = F 1, similarly for F ( ) and F( ). (ii) Let F (ωa) (iii) y definition, let F (ωb) be the second Fermat point of F (ω) the points F (ωb), F(ωb). (iv) y definition, let F (ωc) be a reflection of the point F (ω) be a reflection of the point F (ω), same for F(ωa), F(ωa). wrt side. Similarly define wrt midpoint of. Similarly define the points F (ωc), F(ωc). Then we have that : (1) If ω is odd, then the points F 2, F (ω), F(ω), F(ω) lie on the same circle. (2) If ω is even, then the points F 1, F (ω), F(ω), F(ω) lie on the same circle. Theorem 2.5. Given triangle with the first Fermat point F. Let F be the second Fermat point for F. Similarly define F, F. Let F be the first Fermat point of FF and F be the first Fermat point of FF. Let two tangents from F, F to circle (F F ) meet at X. Similarly define X, X. Then F, X, X, X are collinear Miquel points. Theorem 2.6. onsider any triangle and any point X. Let P = X, Q = X, R = X. Let M Q be the Miquel point of the lines X, X,,, similarly define the points M P, M R. Denote l Q as the Simson line of the point M Q wrt triangle XP (it s same for the triangles XR, P, R). Similarly define the lines l P, l R. Let P Q R be the midpoint triangle for PQR. Let P Q l R = R, like the same define the points P,Q. Then P P, Q Q, R R are concurrent Steiner lines. Theorem 2.7. onsider any triangle and any point P. Let L be the Steiner line for the lines,, P, P, similarly define the lines L, L. Let the lines L, L, L form a triangle. Then the circumcircle of, the pedal circle of P wrt and the nine - point circle of passes through the same point.

3 2.6. Isogonal conjugations. SOME NEW THEOREMS IN PLNE GEOMETRY 3 Theorem 2.8. Let the incircle of touches sides,, at 1, 1, 1 respectively. Let the,, excircles of touches sides,, at 2, 2, 2 respectively. Let 1 be isogonal conjugate to the point wrt Similarly define the points 1 and 1. Let 2 be isogonal conjugate to the point wrt Similarly define the points 2 and 2. Then we have that : (1) Lines 2 1, 2 1, 2 1 are concurrent at X2 1. (2) Lines 1 2, 1 2, 1 2 are concurrent at X1 2. (3) Points 1, 1, 1, 2, 2, 2, X2 1, X2 1 lie on the same conic Deformation principle for circles. If in the particular case of an equilateral triangle some two circles are equivalent, then in the general case the radical line of these two circles has some natural relations with respect to the base triangle. Theorem 2.9. Let given triangle and its centroid G. Let the circumcenters of the triangles G, G, G form a triangle with the circumcircle ω. Then the circumcenter of the pedal triangle of G wrt lies on the radical line of () and ω. Theorem onsider triangle with the first Fermat point F and let O, O, O be the circumcenters of F, F, F respectively. Let X be a reflection of O wrt, similarly define the points X, X. Let l be the radical line of (O O O ) and (). Then F is the Miquel point of the lines X X, X X, X X, l. Remark 2.1. In all theorems from this section consider the case of an equilateral triangle. 3. Nice fact about triangle centers For any i,j N if some fact is true for the Kimberling center X i, then we can try to transport the same construction from the triangle center X i to the center X j and to look on the nice properties of the resulting configuration. Theorem 3.1. Given a triangle. Let excircle tangent to at 1. Similarly define 1, 1. Let be the midpoint triangle of. onsider intersection N of the perpendiculars to,, from 1, 1, 1 respectively. Let the circle ( ) meet sides, second time at 3, 3 respectively. Let X = Then X N. Remark 3.1. onsider the case when N is the incenter of. Theorem 3.2. Let the incircle of touches sides,, at,, respectively. Reflect, wrt line and get the points 1, 1 respectively. Let = 1 1, similarly define the points,. Then the points,,, I, G lie on the same circle, where G is the Gergonne point of. Remark 3.2. onsider the case when,, goes through the first Fermat point. Theorem 3.3. Let the incircle of touches sides,, at,, respectively. Reflect point wrt lines, and get the points, respectively. Let =, similarly define the points,. Then the incenter of coincide with the circumcenter of.

4 4 LEXNDER SKUTIN 4. onstruction of midpoint analog Definition 4.1. For any pairs of points, and, D denote M(,D) as the Miquel point of the complete quadrilateral formed by the four lines, D,, D. Definition 4.2. For any point X and a segment YZ denote M(X,YZ) as a point P, such that the circles (PXY) and (PXZ) are tangent to segments XZ, XY at X. onsider any two segments and D, then the point M(,D) can be seen as midpoint between the two segments, D. lso we can consider the segment and the point and to look on the point M(,) as on the midpoint between the point and the segment. Remark 4.1. In the case when = and = D we will get that the points M(,) and M(,D) are midpoints of. Theorem 4.1. Let given segments P Q, P Q, P Q, P D Q D. Let ω D be the circumcircle of M(P Q,P Q )M(P Q,P Q )M(P Q,P Q ). Like the same define the circles ω, ω, ω. Then the circles ω, ω, ω, ω D intersect at the same point. Theorem 4.2 (Three nine-point circles intersect). Let given circle ω and points P, Q, P, Q, P, Q on it. Let = M(P Q,P Q ), similarly define points,. onsider points 1 = P P Q Q, 1 = P P Q Q, 1 = P P Q Q. Let the lines 1, 1, 1 form a triangle with the nine - point circle ω 1. Then the circles ω 1, ( ), ( ) pass through the same point. Theorem 4.3. onsider triangle. Let points P, Q considered such that PP QQ, where P, P are isogonal conjugated and Q, Q are isogonal conjugated wrt triangle. Let P be the second intersection point of line P with circle (P ). Similarly define the points P, P. Let Q be the second intersection point of line Q with circle (Q ). Similarly define the points Q, Q. Let Q P Q P = R, similarly Q P Q P = R and Q P Q P = R. Then we have that : (1) ircles (M(Q P,Q P )M(Q P,Q P )M(Q P,Q P )) (M(Q P,Q P )M(Q P,QP)M(QP,Q P )) and (R R R ) goes through the same point. (2) Lines R, R, R intersect at the same point which lies on the circle (M(Q P,QP)M(Q P,QP)M(QP,Q P )) Remark 4.2. onsider the case when P = Q incenter of. Theorem 4.4. Let P and Q be two isogonal conjugated points wrt. Let O P, OP, O P be the circumcenters of the triangles P, P, P respectively. Similarly let O Q, OQ, OQ be the circumcenters of the triangles Q, Q, Q respectively. Let M = M(O POQ,OP OQ ), similarly define the points M, M. Let O be the circumcenter of. Then we have that : (1) ircles (OM ), (OM ), (OM ) are coaxial. (2) Let = M(PQ,O P OQ ), = M(PQ,O P OQ ) and = M(PQ,O P OQ ). Then lies on (OM ). (3) Points,,, M, M, M lie on the same circle.

5 SOME NEW THEOREMS IN PLNE GEOMETRY 5 5. Deformation of segment into a conic 5.1. Deformation principle for conics. In some statements we can replace some segment by a conic. Theorem 5.1. onsider a circle ω and the two conics K 1, K 2 which are tangent to ω at four points P 1, Q 1, P 2, Q 2. Where P 1 and Q 1 lie on K 1, P 2 and Q 2 lie on K 2. onsider two external tangents to K 1, K 2 which intersect at the point. Let the circles (P 1 Q 1 ) and (P 2 Q 2 ) intersect second time at E. Then EO = π/2, where O is the center of ω. Theorem 5.2. onsider a circle ω and the two conics K 1, K 2 which are tangent to ω at four points. Let K 1 has foci F 1, F 2 and K 2 has foci F 3, F 4. Let l 1, l 2 be two external tangents to K 1, K 2. Let Y = F 1 F 4 F 2 F 3 and the circles (F 1 YF 3 ), (F 2 YF 4 ) meet second time at X. onsider the point Z = l 1 l 2 (see picture below). Then X, Y, Z are collinear. ω F 1 K 1 X Y F 3 K2 Z F 4 F 2 Definition 5.1. For any two conics 1 and 2 consider all their four tangents l 1, l 2, l 3, l 4. y definition let M( 1, 2 ) be the Miquel point for the lines l 1, l 2, l 3, l 4. Theorem 5.3. Let given conics,,, D. Let ω D be the circumcircle of triangle M(, )M(, )M(, ) Like the same define ω, ω, ω. Then ω, ω, ω, ω D intersect at the same point. Theorem 5.4. onsider any triangle. Let P, P and Q, Q be two pairs of isogonal conjugated points wrt. onsider a conic P with foci at P, P and also consider a conic Q with foci at Q, Q. Then the points,,, M( P, Q ) lie on the same circle.

6 6 LEXNDER SKUTIN 6. On incircles in orthocenter construction Main idea of this section is to construct something nice which includes incircles in orthocenter construction. Theorem 6.1. onsider triangle with orthocenter H and altitudes,,. Let =. Let I 1, I 2, I 3, I 4, I 5, I 6 be the incenters of the triangles,, H, H, H, H respectively. Then we have that : (1) Lines I 1 I 4, I 2 I 3, are concurrent. (2) Lines I 6 I 4, I 5 I 3, are concurrent. Theorem 6.2. Let given triangle with orthocenter H and altitudes H, H, H. Let the second tangent line through H to excircle of meet the second tangent line through H to excircle of at 1. Let the second tangent line through H to excircle of meet the second tangent line through H to excircle of at 2. Then H, 1, 2 are collinear. Theorem 6.3. Let given triangle with altitudes H, H, H. Let l be the line which passes through the incenters of H and H. Similarly define l, l. Then the triangle formed by l, l, l is perspective to. Theorem 6.4. Let I be the incenter of. Let H a be the orthocenter of I, similarly define H b, H c. Let I a be the incenter of H a, similarly define I b and I c. Then (1) Point I is the orthocenter of I a I b I c. (2) Let H a (2) be the orthocenter of I a, similarly define H (2) b, H c (2). Then the triangles H a (2) H (2) b H c (2) and are perspective. (3) Let I (2) a be the incenter of H (2) a, similarly define I (2) b, I (2) and I a (2) I (2) b I c (2) are perspective. c. Then the triangles Theorem 6.5 (Four nine-point circles intersect). Let given triangle with orthocenter H. Let ω be the incircle of and ω be the incircle of H. Like the same define the circles ω, ω. Let l be the second external tangent line of the circles ω, ω. Similarly define the lines l, l. Let = l l, = l l, = l l. onsider the point! = l. Similarly define the points!,!. Let π be the pedal circle of H wrt. Then the nine - point circles of the triangles,, circle π and circle (!!! ) meet at the same point. Theorem 6.6. Let given triangle with orthocenter H. Let,, be altitudes of. Let I 1, I 2 be the incenters of the triangles, respectively. Let I 1, I 2 be excenter of the triangle and respectively excenter of the triangle. Then the points I 1, I 2, I 1, I 2,, lie on the same conic. Theorem 6.7. Let,, be altitudes of. Let H be the orthocenter of and be the midpoint triangle of. Places of the points,, wrt points 1, 1, 1 are the same as on the picture below. Let the incenters of the triangles 1 H, 1 H and H excenter of H 1 lie on the circle ω 1. Let H excenters of the triangles 1 H, 1 H and the incenter of H 1 lie on the circle ω 2. Then (1) Point H lies on the radical line of ω 1 and ( ). (2) ircles ω 1 and ω 2 are symmetric wrt Euler line of. (3) ircles ω 1, ω 2, ( ) are coaxial.

7 SOME NEW THEOREMS IN PLNE GEOMETRY H 1 Remark 6.1. ompare previous theorem with ideas of Section nalog for cyclic quadrilateral. Theorem 6.8. Let given convex cyclic quadrilateral D, where D. Let D = H. Let X be the foot of perpendicular from H on. Let Y be the foot of perpendicular from H on D. Let M and N be the midpoints of segments and D respectively. Let I 1, I 2, I 3, I 4 be the incenters of the triangles HXM, D, HYN, D respectively. Let W be the midpoint of a smaller arc D of a circle (D). Then I 1 I 2, I 3 I 4 and HW are concurrent.

8 8 LEXNDER SKUTIN N D Y I 3 I 4 W H I 2 I 1 X M 6.2. Tangent circles. Theorem 6.9. Let given triangle with orthocenter H. Let,, be altitudes of. Let the circle ω 1 is internally tangent to the incircles of the triangles H, H, H. Similarly let the circle ω 2 is internally tangent to the incircles of the triangles H, H, H. Then H lies on the radical line of ω 1 and ω 2. H

9 SOME NEW THEOREMS IN PLNE GEOMETRY 9 Theorem Let given triangle with orthocenter H. Let W, W, W be the reflections of H wrt lines,, respectively. Let I be H excenter of the triangle HW (and let (I ) be the excircle itself), I be H excenter of the triangle HW, I be H excenter of the triangle HW, I be H excenter of the triangle HW, I be H excenter of the triangle HW, I be H excenter of the triangle HW. Suppose that the circle ω 1 externally touches to (I ), (I ), (I ). Suppose that the circle ω 2 externally touches to (I ), (I ), (I ). Then we have that : (1) Point H lies on the radical line of (I I I ) and (I I I ). (2) Point H lies on the radical line of π 1 and π 2. I I W W I H I W I I

10 10 LEXNDER SKUTIN W W H W 6.3. onjugations. Note that isogonal conjugation can be considered as a conjugation associated with the incenter. Theorem Let I be the incenter of. Let be the cevian triangle of I wrt. Let H I be the orthocenter of I. Similarly define the points H I, H I, H I, H I, H I. Let i I be isogonal conjugation of H I wrt, similarly define the points i I, i I, i I, i I, i I. Then i I i I, i I i I, i I i I are concurrent. Theorem Let given triangle with orthocenter H. Let be the reflection of H wrt, similarly define the points,. Let =, =. Like the same define the points,,,. Let i (H) be isogonal conjugation of H wrt. Similarly define the points i (H), i (H), i (H), i (H), i (H). Then we have that : (1) Lines i (H)i (H), i (H)i (H), i (H)i (H) are concurrent. (2) Let i (H) be isogonal conjugation of H wrt, similarly define the points i (H), i (H). Then the point H is the orthocenter of the triangle i (H)i (H)i (H).

11 SOME NEW THEOREMS IN PLNE GEOMETRY ombination of different facts Here we will build combinations of some well-known constructions from geometry. Theorem 7.1 (Pappus and eva). onsider triangles and and arbitrary points P, Q. Let be the cevian triangle of P wrt and let be the cevian triangle of Q wrt Let R = , S = and T = Then by the Pappus s theorem points R, S, T lie on the same line l. Similarly define the lines l, l. Then the triangle formed by the lines l, l, l is perspective to the both triangles and Theorem 7.2 (Two types of tangency). Let given triangle and let circle ω is tangent to the lines, and externally tangent to the circle (), similarly define the circles ω and ω. Let the circle Ω goes through the points, and internally tangent to the incircle of, similarly define the circles Ω and Ω. Let h be the external homothety center of the circles ω and ω, let H be the external homothety center of the circles Ω and Ω. Let the line L goes through h and H. Similarly define the lines L and L. Let the lines L, L, L form a triangle H H H. Let H be the external homothety center of the circles ω and Ω, similarly define H, H. Then the triangles H H H and H H H are perspective at the incenter of. Theorem 7.3 (onic and Poncelet point). onsider a conic and any point X. Let given points 1, 1, 1, D 1, 2, 2, 2, D 2 on, such that the lines 1 2, 1 2, 1 2, D 1 D 2 goes through the point X. onsider the circle Ω which goes through the Poncelet points of quadrilaterals , , Similarly define the circles Ω D, Ω D, Ω D. Then all these circles Ω, Ω D, Ω D, Ω D goes through the same point. Theorem 7.4 (onic and Miquel point). onsider a conic and any point X. Let given points 1, 1, 1, D 1, 2, 2, 2, D 2 on, such that the lines 1 2, 1 2, 1 2, D 1 D 2 goes through the point X. onsider the circumcircle Ω of the triangle M( 1 2, 1 2 )M( 1 2, 1 2 )M( 1 2, 1 2 ) (see definition of M in Section 4). Similarly define a circle Ω D. Then we get that Ω D = Ω ombinations with radical lines. Theorem 7.5 (Gauss line theorem and radical lines). onsider any four lines l 1, l 2, l 3, l 4 and let l be the Gauss line for these four lines. onsider any four points P 1 l 1, P 2 l 2, P 3 l 3, P 4 l 4. Let X ij = l i l j, for any index i j. Let r 1 be the radical line of circles (P 1 P 2 X 12 ) and (P 3 P 4 X 34 ), let r 2 be the radical line of circles (P 1 P 3 X 13 ) and (P 2 P 4 X 24 ), finally let r 3 be the radical line of circles (P 1 P 4 X 14 ) and (P 2 P 3 X 23 ). onsider the case when l 1, l 2, l 3 intersect at the same point P. Then we get that P lies on l. Theorem 7.6 (Pascal s theorem and radical lines). onsider points,,, D, E, F which lie on the same circle. Let X = E F, Y = D E. Let l 1 be the radical line of circles (EFX) and (Y), let l 2 be the radical line of circles (DYE) and (X), and let l 3 be the radical line of circles (XF) and (DY). onsider the Pascal line L of hexagon EFD (i.e L = XY ). Let the line W goes through the center of (DEF) and is perpendicular to L. Then we have that : (1) Lines W, E, l 3 are concurrent. (2) Lines W, l 1, l 2 are concurrent.

12 12 LEXNDER SKUTIN Theorem 7.7 (Morley s theorem and radical lines). onsider any triangle and it s Morley s triangle. Let l be the radical line of ( ) and ( ). Similarly define the lines l, l. Then the lines l, l, l form an equilateral triangle which is perspective to. Theorem 7.8 (Napoleon s theorem and radical lines). onsider any triangle and it s Napoleon s triangle. Let l be the radical line of ( ) and ( ). Similarly define the lines l, l. Then the lines l, l, l are concurrent On IMO 2011 Problem 6. Here we will represent some interesting facts which are related to the following IMO 2011 Problem 6 [5, Problem G8]. Definition 7.1. For any triangle and any point P on (), denote ω as the circumcircle of triangle formed by the reflections of the tangent line through P to () wrt sides of. y definition let (,P) be a tangent point of ω and (). Theorem 7.9. For any four lines l 1, l 2, l 3, l 4 consider triangles ijk, which are formed by the lines l i, l j, l k. Let M be the Miquel point of the lines l 1, l 2, l 3, l 4. Then the points ( 123,M), ( 124,M), ( 234,M), ( 134,M) lie on the circle (l 1,l 2,l 3,l 4 ). Theorem Let given a parabola K and lines l 1, l 2, l 3, l 4, l 5 which are tangent to K. Then the circles (l 1,l 2,l 3,l 4 ), (l 1,l 2,l 3,l 5 ), (l 1,l 2,l 5,l 4 ) goes through the same point. 8. low-ups Theorem 8.1 (low-up of Desargues s theorem). onsider circles ω, ω, ω, ω, ω, ω with centers at points,,,,, respectively. Let H be the external homothety center of the circles ω, ω. Similarly define the points H, H, H, H, H, H, H, H, H. Let the triangle H formed by the lines H H, H H, H H. Then we have that : (1) Triangle H is perspective to H H H, to and to. (2) ll these perspective centers lie on the same line. Theorem 8.2. onsider points,,, D and let P = D and Q = D. onsider segments 1 2, 1 2, 1 2 and D 1 D 2. Let given that the segments 1 2, 1 2, 1 2, D 1 D 2 are parallel to each other and that the points,,, D are the midpoints of the segments 1 2, 1 2, 1 2, D 1 D 2 respectively. Let also given that P = D 1 = D 2 and Q = D 1 = D 2. Let the circle Ω P goes through the points M(P, 1 2 ), M(P, 1 2 ), M( 1 2, 1 2 ) and let the line L P goes through the center of Ω P and is perpendicular to D (see definition of M in Section 4). Similarly define the lines L PD, L Q, L QD. onsider the case when L P, L PD, L Q intersect at the same point W. Then L QD also goes through W. Theorem 8.3 (low-up of Miquel point). onsider any four points,,, D. Let P = D, Q = D. onsider circles ω, ω, ω, ω D with centers at points,,, D and radii r, r, r, r D respectively. Let given that P Q Q = r r, QD Q = r D P = r r, P PD = r rd, r. Let the circle Ω P goes through P and internally touches the circles ω, ω, let the circle Ω Q goes through Q and internally touches the circles ω, ω. Similarly define the circles Ω QD and Ω PD. Then there exists a circle ω M which is tangent to Ω P, Ω Q, Ω QD and Ω PD.

13 SOME NEW THEOREMS IN PLNE GEOMETRY 13 Theorem 8.4 (low-up of Pascal s theorem). onsider two circles Ω 1 and Ω 2. Let circles ω, ω, ω, ω D, ω E, ω F are tangent to both Ω 1 and Ω 2. onsider two external tangents L (1), L(2) to the circles ω, ω, similarly consider two external tangents L (1) D, L (2) D to the circles ω, ω D. Let the lines L (1), L(2), L(1) D, L(2) D form a convex quadrilateral,d and let P,D be the intersection of the diagonals of,d. Similarly define,de, D,EF and the points P,DE, P D,EF. Then P,D, P,DE, P D,EF are collinear. Theorem 8.5 (low-up of rianchon s theorem). onsider two circles Ω 1 and Ω 2. onsider two circles ω, ω which tangent to both Ω 1 and Ω 2 and also tangent to each other. Similarly define pairs of circles ω, ω ; ω, ω ; ω D, ω D ; ω E, ω E ; ω F, ω F. onsider two external tangents L (1), L(2) to the circles ω, ω, similarly consider two external tangents L (1), L(2) to the circles ω, ω. Let the lines L(1), L(2), L(1), L(2) form a convex quadrilateral, and let P, be the intersection of the diagonals of,. Similarly define,,,d, D,E, E,F, F, and P,, P,D, P D,E, P E,F, P F,. Then P, P D,E, P, P E,F, P,D P F, are concurrent. 9. Generalizations of Feuerbach s theorem 9.1. The Feuerbach point for a set of coaxial circles. onsider any triangle and let given three circles ω, ω, ω, such that ω is tangent to, ω is tangent to, ω is tangent to and also given that the circles ω, ω, ω, () are coaxial. Then the set of circles ω, ω, ω can be seen as a generalized incircle for. Theorem 9.1. onsider any triangle and let given three circles ω, ω, ω (with centers at O, O, O respectively), such that ω is tangent to at, ω is tangent to at, ω is tangent to at and also given that the circles ω, ω, ω, () are coaxial. Then we have that : (1) (Gergonne point) Lines,, are concurrent. (2) (Feuerbach point) Pedal circles of O, O, O wrt, the nine - point circle of and the circle ( ) goes through the same point The Feuerbach point for a set of three conics. We can look on the set of three confocal conics (same situation as [1, Problem 11.18]) as on the generalized circle. lso, we can look at a construction with three conics and their three tangents (see picture below), as on a generalized incircle construction.

14 14 LEXNDER SKUTIN Theorem 9.2. onsider points,,. Let given set of conics K, K, K, where conic K has foci and, similarly conic K has foci and and conic K has foci and. onsider the tangent line l to conics K, K, similarly define the lines l, l. Let the lines l, l, l form a triangle. Let K be the foot of perpendicular from to l. Similarly define K and K. Then (1) (Gergonne point) Triangles and K K K are perspective. (2) (nother incenter) Lines,, are concurrent at I. (3) (Feuerbach point) Pedal circle of point I wrt, circle (K K K ), the nine - point circle of and the cevian circle of I wrt goes through the same point. L L evian circle I Pedal circle K K L K

15 SOME NEW THEOREMS IN PLNE GEOMETRY Isogonal conjugacy for isogonal lines onsider any situation with a triangle and pointsp,q, such that P = Q. Next we will see that it s natural to consider isogonal conjugations of P, Q wrt. Theorem Let the incircle of is tangent to the sides of at Let excircle of is tangent to the sides of at Let 1 1 be 1 altitude of Similarly let 2 2 be 2 altitude of onsider isogonal conjugations, of, wrt 1 2 respectively. Then the midpoint of lies on. Theorem onsider any four points,,, D. Let P = D, Q = D. onsider a point N, such that DNQ = PN. onsider isogonal conjugations 1, 2 of wrt triangles ND and NPQ respectively. onsider isogonal conjugations Q 1, Q 2 of Q wrt triangles N and ND respectively. onsider isogonal conjugations D 1, D 2 of D wrt triangles N and NPQ respectively. Then all the points 1, 2, Q 1, Q 2, D 1, D 2 lie on the same conic. Theorem onsider any triangle and any two points P, Q on it s circumcircle. onsider a point on, such that is the external angle bisector of P Q. Similarly define on and on. onsider isogonal conjugations, of, wrt P Q respectively. Similarly define the points,,,. Then all the points,,,,, lie on the same conic. Remark Note that the previous three theorems are related to [1, Problem 4.5.7] to [1, Problem ] and to [1, Problem 3.13]. 11. Fun with some lines In this section we will construct some nice theorems which includes some famous lines. Theorem 11.1 (Gauss lines). onsider any four lines l 1, l 2, l 3, l 4. Let l be the Gauss line of the complete quadrilateral formed by these four lines. Let g 1 be the Gauss line of the complete quadrilateral formed by four lines l, l 2, l 3, l 4. Similarly define the lines g 2, g 3, g 4. Then the lines l, g 1, g 2, g 3, g 4 are concurrent. Theorem 11.2 (Steiner lines). onsider any four lines l 1, l 2, l 3, l 4. Let l be the Steiner line of complete quadrilateral formed by these four lines. Let g 1 be the Steiner line of the complete quadrilateral formed by four lines l, l 2, l 3, l 4. Similarly define the lines g 2, g 3, g 4. Then the Steiner line of the complete quadrilateral formed by g 1, g 2, g 3, g 4 is parallel to l. Theorem 11.3 (Euler lines). Let given triangle, let the Euler line of cut it s sides at 1, 1, 1 respectively. So we get three new triangles 1 1, 1 1, 1 1. Let three Euler lines of 1 1, 1 1, 1 1 form a triangle. Then we have that : (1) Euler lines of and are the same. (2) Let the Euler line of and three side lines of form triangles, 1, 2, 3. Then the Euler lines of 1, 2, 3 form a triangle. Theorem 11.4 (Simson lines). Let given triangle, let P be any point on it s circumcircle. Let the Simson line of P wrt intersects with () at Q, R. Let the Simson lines of Q, R wrt intersect at N. Then the middle of PN lies on QR.

16 16 LEXNDER SKUTIN 12. On Three Pascal lines onsider some nice situation which includes three Pascal lines l 1,l 2, l 3 of some nice hexagons. Then we can predict that these lines are concurrent at the same point. Theorem Let given triangle with orthocenter H. Let,, be altitudes of. Let I 1, I 2 be the incenters of the triangles, respectively. Let I 1, I 2 be excenter of and respectively excenter of. From theorem 6.6 we know that the points I 1, I 2, I 1, I 2,, lie on the same conic, so let l be the Pascal line of hexagon I 1 I 1 I 2I 2 in this order. Similarly define the lines l, l. Then l, l, l are concurrent. Theorem onsider triangle and it s first Fermat point F. Let O be the circumcenter of F. Similarly define the points O, O. Let X be the reflection of O wrt side. Similarly define the points X, X. It s well known that circles (O O ), (O O ) and (O O ) intersect at the second Fermat point F 2. Let the lines F 2 X, F 2 X intersect with (X X ) second time at T, T respectively. Similarly define the points T, T, T, T. Let l be the Pascal line of hexagon T T O O in this order. Similarly define l, l. Then l, l, l are concurrent. Theorem onsider any points,, on the sides,, of respectively. Let the circles ( ), ( ) and ( ) intersect at X. Let a line through point X and perpendicular to intersects with ( ) at P and a line through point X and perpendicular to intersects with ( ) at P. Similarly define the points P, P on ( ), and P, P on ( ). Let l be the Pascal line of hexagon P XP in this order. Similarly define l, l. Then we have that : (1) Lines, l, XP are concurrent. (2) Let = l l, = l l, = l l. Then XP,, are concurrent New Pascal line. onsider any conic and let conics 1, 2, 3 are tangent to it at six points. Let the common internal tangents to the conics 1, 2 meet at X 12. Let the common internal tangents to conics 1, 3 meet at X 13. Let the common internal tangents to the conics 2, 3 meet at X 23. Then X 12, X 23, X 13 are collinear. See picture below for more details. X 23 X 13 X Definition Name this line as Pasc ( 1, 2, 3 ), where order of conics is important. Theorem onsider three conics,,. Let given that there exists a conic, which is tangent to them at six points. Let two outer tangents to the conics, intersect with two outer tangents to the conics, at the points P (1), P(2), P(3), P(4). Similarly define P (1), P(2), P(3), P(4), P(1), P(2), P(3), P(4) (see picture below for more details). Let by definition union of the external tangents to conics, form a conic. Similarly define conics,. Note that we can look on the segments XY as on the degenerated conics. Let the lines Pasc (,P (1) P(4), ) and Pasc (,P (1) P(2), ) meet at W. Similarly define W, W. Then P (1) W, P (1) W, P (1) W are concurrent.

17 SOME NEW THEOREMS IN PLNE GEOMETRY 17 P (3) P (4) P (2) P (3) W P (1) W P (2) P (1) P (1) W P (4) P (4) P (2) P (3) 13. Facts related to the set of confocal conics Theorem onsider any set of points 1, 2,..., n. Let given a set of conics K 1, K 2,..., K n, where the conic K i has foci i and i+1, for every 1 i n ( n+1 = 1 ). For each integer i, consider the line l i passing through two intersection points of the conics K i, K i+1. Then in the case when the lines l 1, l 2,..., l n 1 pass through the same point P we get that the line l n also passes through P. Remark Previous theorem can be seen as n - conic analog of [1, Problem 11.18]. y observing theorem 13.1 one can note that P can be seen as the incenter for the set of conics K 1, K 2,..., K n. So if we apply this analogy to [1, Problem ], then we ll obtain the following theorem. Theorem onsider cyclic quadrilateral D. Let K be an ellipse with foci and. Let K D be an ellipse with foci and D. Let the segments, D,, D intersect with the ellipse K at T, T D, T, T D respectively. Let the segments,, D, D intersect with the ellipse K D at T, T D, T D, T D respectively. onsider the intersection point I of the tangent lines through T and T to K. onsider the intersection point I D of the tangent lines through T D and T D to K. Similarly define the points I, I (similarly for ellipse K D ). Then the points I, I, I, I D lie on the same circle.

18 18 LEXNDER SKUTIN We can look at the set of three confocal conics (same situation as [1, Problem 11.18]) as on the analog of the situation where given three circles ω, ω, ω with centers at, and. So [1, Problem 11.18] can be seen as an analog of radical center theorem. Next two pictures describes conic analogs of [1, Theorem 6.3.7] and [1, Theorem 10.11]. Definition onsider any two conics K 1 and K 2 which share same focus F. Let the conics K 1 and K 2 intersect at the points,. onsider the intersection point X of the tangents to the conic K 1 from,. Similarly let Y be the intersection point of the tangents to the conic K 2 from,. Then let by definition L F (K 1,K 2 ) = XY. Theorem onsider points,,. Let given set of conics K, K, K, where conic K has foci and, similarly for K, K. Let X be the intersection point of the lines L (K,K ) and L (K,K ). Similarly define the points X and X. Then X, X and X are concurrent. Theorem onsider cyclic quadrilateral D. Let P = D. Let conic K has foci at points, and goes through the incenter of P. Like the same let conic K has foci at points, and goes through the incenter of P. Similarly define conics K D and K D. onsider P as the intersection point of the lines L (K,K D ) and L (K,K ). Similarly define the points P, P D, P D. Then we have that : (1) Lines P P D, P P D intersect at P. (2) Define point as the second intersection point of the line L (K,K D ) with (D). Similarly define the points,, D. Then the lines and D intersect at P.

19 SOME NEW THEOREMS IN PLNE GEOMETRY 19 References [1]. V. kopyan, Geometry in Figures, reatespace, [2]. Kimberling, Encyclopedia of Triangle enters - ET, [3]. Skutin, On rotation of a isogonal point. Journal of classical geometry, vol 2, 66-67, [4]. V. kopyan and.. Zaslavsky, Geometry of conics. merican Mathematical Society, New York, 2007 [5] International Mathematical Olympiad, [6] P. Yiu, Introduction to the geometry of the triangle, [7] I. ogdanov, Two theorems on the focus-sharing ellipses: a three-dimensional view. Journal of classical geometry, vol 1, 1-5, [8] F. Nilov, generalization of the Dandelin theorem. Journal of classical geometry, vol 1, 57-65, 2012.