A Theorem about Simultaneous Orthological and Homological Triangles

Transcription

1 Theorem about Simultaneous Orthological and Homological Triangles Ion Pătraşcu Frații uzești ollege, raiova, Romania Florentin Smarandache University of New Mexico, Gallup ampus, US bstract. In this paper we prove that if P, P are isogonal points in the triangle, and if and are their ponder triangle such that the triangles and are homological (the lines,, are concurrent), then the triangles and are also homological. Introduction. In order for the paper to be self-contained, we recall below the main definitions and theorems needed in solving this theorem. lso, we introduce the notion of Orthohomological Triangle, which means a triangle that is simultaneously orthological and homological. Definition In a triangle the evians and which are symmetric with respect to the angle s bisector are called isogonal evians. Fig. Observation If, and, are isogonal evians then. (See Fig..) Theorem (Steiner)

2 then: If in the triangle, and are isogonal evians,, are points on = We have: sin ( ) areaδ = = () areaδ sin ( ) sin ( ) areaδ = = () areaδ sin ( ) ecause sin ( ) = sin ( ) and sin ( ) = sin ( ) by multiplying the relations () and () side by side we obtain the Steiner relation: = (3) Theorem In a given triangle, the isogonal evians of the concurrent evians are concurrent. We ll use the eva s theorem which states that the triangle s evians,, (,, ) are concurrent if and only if the following relation takes place: = (4) P P Fig.

3 We suppose that,, are concurrent evians in the point P and we ll prove that their isogonal,, are concurrent in the point P. (See Fig. ). From the relations (3) and (4) we find: = (5) = (6) = (7) y multiplying side by side the relations (5), (6) and (7) and taking into account the relation (4) we obtain: =, which along with eva s theorem proves the proposed intersection. Definition The intersection point of certain evians and the point of intersection of their isogonal evians are called isogonal conjugated points or isogonal points. Observation The points P and P from Fig. are isogonal conjugated points. In a non right triangle its orthocenter and the circumscribed circle s center are isogonal points. Definition 3 If P is a point in the plane of the triangle, which is not on the triangle s circumscribed circle, and ', ', ' are the orthogonal projections of the point P respectively on,, and, we call the triangle ' ' ' the podaire triangle of the point P. Definition 4 The podaire triangle of the center of the inscribed circle in the triangle is called the contact triangle of the given triangle. 3

4 F Fig. 3 Observation 3 In figure 3, ' ' ' is the contact triangle of the triangle. The name is connected to the fact that its vertexes are the contact points (of tangency) with the sides of the inscribed circle in the triangle. Definition 5 The podaire triangle of the orthocenter of a triangle is called orthic triangle. Definition 6 Two triangles are called orthological if the perpendiculars constructed from the vertexes of one of the triangle on the sides of the other triangle are concurrent. Definition 7 The intersection point of the perpendiculars constructed from the vertexes of a triangle on the sides of another triangle (the triangles being orthological) is called the triangles orthology center. Theorem 3 (The Orthological Triangles Theorem) If the triangles and ' ' ' are such that the perpendiculars constructed from on, ' ' from on ' ' and from on ' ' are concurrent (the triangles and ' ' ' being orthological), then the perpendiculars constructed from ' on, from ' on, and from ' on are also concurrent. To prove this theorem firstly will prove the following: Lemma (arnot) If is a triangle and,, are points on,, respectively, then the perpendiculars constructed from on, from on and from on are concurrent if and only if the following relation takes place: + + = (8) 0 4

5 M If the perpendiculars in,, are concurrent in the point M (see Fig. 4), then from Pythagoras theorem applied in the formed right triangles we find: Fig. 4 hence Similarly it results = M M (9) = M M (0) = M M () = M M () = M M (3) y adding these relations side by side it results the relation (8). Reciprocally We suppose that relation (8) is verified, and let s consider the point M being the intersection of the perpendiculars constructed in on and in on. We also note with ' the projection of M on. We have that: ' ' = 0 (4) omparing (8) and (4) we find that = ' ' and ( )( + ) = ( ' ' )( ' + ' ) and because = ' + ' = we obtain that ' =, therefore the perpendicular in passes through M also. Observation 4 The triangle and the podaire triangle of a point from its plane are orthological triangles. 5

6 The proof of Theorem 3 Let s consider and ' ' ' two orthological triangles (see Fig. 5). We note with M the intersection of the perpendiculars constructed from on, ' ' from on ' ' and from on ' ', also we ll note with,, the intersections of these perpendiculars with ' ', and ' ' ' ' respectively. M M Fig. 5 In conformity with lemma, we have: ' ' + ' ' + ' ' = 0 (5) From this relation using the Pythagoras theorem we obtain: ' ' + ' ' + ' ' = 0 (6) ' ' ' We note with,, the orthogonal projections of ', ', ' respectively on,,. From the Pythagoras theorem and the relation (6) we obtain: from ' ' ' ' ' ' + + = 0 (7) This relation along with Lemma shows that the perpendiculars drawn from ' on, ' on and from ' on are concurrent in the point M '. The point M ' is also an orthological center of triangles ' ' ' and. Definition 8 The triangles and ' ' ' are called bylogical if they are orthological and they have the same orthological center. Definition 9 Two triangles and ' ' ' are called homological if the lines ', ', ' are concurrent. Their intersection point is called the homology point of triangles and ' ' ' 6

7 Observation 6 In figure 6 the triangles ', ', ' are homological and the homology point being O O Fig.6 If is a triangle and ' is ' ' its podaire triangle, then the triangles and are ' ' ' homological and the homology center is the orthocenter H of the triangle Definition 0 number of n points ( n 3) are called concyclic if there exist a circle that contains all of these points. Theorem 5 (The circle of 6 points) If is a triangle, P, Pare isogonal points on its interior and respectively P P P Fig. 7 7

8 the podaire triangles of P and P, then the points,,,,, are concyclic. We will prove that the 6 points are concyclic by showing that these are at the same distance of the middle point P of the line segment P P. It is obvious that the medians of the segments ( ),( ),( ) pass through the point P, which is the middle of the segment ( PP ).The trapezoid PP is right angle and the mediator of the segment ( ) will be the middle line, therefore it will pass through P,(see Fig. 7). Therefore we have: P = P, P = P, P = P (8) We ll prove that P = P by computing the length of these segments using the median s theorem applied in the triangles PP and P P. We have: ( ) 4P = P + P PP (9) We note P = x, P = x, m( P) = m( P) = α. In the right triangle P applying the Pythagoras theorem we obtain: P = P + (0) From the right triangle P we obtain: P = Psinα = xsinα and = x cosα From the right triangle P it results = P cos ( α ), therefore = x cos( α ) and P = x sin ( α ), thus = = xcosα xcos( α ) () Substituting back in relation (7), we obtain: P ( ) = x sin α + xcosα xcos α () From the relation (6), it results: 4P = x + x xxcosαcos( α) PP (3) The median s theorem in the triangle P P will give: 4P = ( P + P ) PP (4) ecause P = x sinα, = x cosα, = x cos ( α ), P = P +, we find that 4P = x + x xxcosαcos( α) PP (5) The relations (3) and (5) show that P = P (6) 8

9 Using the same method we find that : P = P (7) The relations (8), (6) and (7) imply that: P = P = P = P = P = P From which we can conclude that,,,,, are concyclic. Lemma (The power of an exterior point with respect to a circle) If the point is exterior to circle Or (, ) and d, dare two secants constructed from that intersect the circle in the points, respectively ED,, then: = E D = cons. (8) The triangles D and E are similar triangles (they have each two congruent angles respectively), it results: D = E T D d E O d Fig. 8 and from here: = E D (9) We construct the tangent from to circle Or (, ) (see Fig. 8). The triangles TE and DT are similar (the angles from the vertex are common and TE DT = m( TE) ). We have: E T =, T D it results E D= T (30) y noting O= a, from the right triangle TO (the radius is perpendicular on the tangent in the contact point), we find that: T = O OT, therefore T = a r = const. (3) The relations (9), (30) and (3) are conducive to relation (8). 9

10 Theorem 6 (Terquem) If,, are concurrent evians in the triangle and,, are intersections of the circle circumscribed to the triangle,, cu ( ), ( ), ( ), then the lines,, are concurrent. Let s consider F the concurrence point of the evians,,. From eva s theorem it results that: = (3) F F Fig 9 onsidering the vertexes,, s power with respect to the circle circumscribed to the triangle, we obtain the following relations: = (33) = (34) = (35) Multiplying these relations side by side and taking into consideration the relation (3), we obtain = (36) This relation can be written under the following equivalent format = (37) From eva s theorem and the relation (37) we obtain that the lines,, are concurrent in a point noted in figure 9 by F. Note The points F and F have been named the Terquem s points by andido of Pisa

11 For example in a non right triangle the orthocenter H and the center of the circumscribed circle O are Terquem s points. Definition Two triangles are called orthohomological if they are in the simultaneously orthological and homological. Theorem 7 (Smarandache-Pătraşcu Theorem of Orthohomological Triangles) If P, P are two conjugated isogonal points in the triangle, and and are their respectively podaire triangles such that the triangles and are homological, then the triangles and are also homological. Let s consider that F is the concurrence point of the evians,, (the center of homology of the triangles and ). In conformity with Theorem 6 the circumscribed circle to triangle intersects the sides ( ), ( ), ( ) in the points,,, these points are exactly the vertexes of the podaire triangle of P, because if two circles have in common three points, then the two circles coincide; practically, the circle circumscribed to the triangle is the circle of the 6 points (Theorem 5). Terquem s theorem implies the fact that the triangles and are homological. Their homological center is F, the second Terquem s point of the triangle. Observation 7 If the points P and P isogonal conjugated in the triangle coincide, then the triangles and, the podaire of P = P are homological. From P = P and the fact that P, P are isogonal conjugate, it results that P = P = I - the center of the inscribed circle in the triangle. The podaire triangle of I is the contact triangle. In this case the lines,, are concurrent in Γ, Gergonne s point, which is the homological center of these triangles. Observation 8 The reciprocal of Theorem 7 for orthohomological triangles is not true. To prove this will present a counterexample in which the triangle and the podaire triangles, of the points P and P are homological, but the points P and P are not isogonal conjugated; for this we need several results. Definition In a triangle two points on one of its side and symmetric with respect to its middle are called isometrics.

12 Definition 3 The circle tangent to a side of a triangle and to the other two sides extensions of the triangle is called exterior inscribed circle to the triangle. Observation 9 In figure 0 we constructed the extended circle tangent to the side ( ). We note its center with I a. triangle has, in general, three exinscribed circles Definition 4 The triangle determined by the contact points with the sides (of a triangle) of the exinscribed circle is called the cotangent triangle of the given triangle. I D D a I a Fig. 0 Theorem 8 The isometric evians of the concurrent evians are concurrent. The proof of this theorem results from the definition 4 and eva s theorem Definition 5 The contact points of the evians and of their isometric evians are called conjugated isotomic points. Lemma 3 In a triangle the contact points with a side of the inscribed circle and of the exinscribed circle are isotomic points. The proof of this lemma can be done computational, therefore using the tangents property constructed from an exterior point to a circle to be equal, we compute the D and (see Fig. 0) in function of the length abc,, of the sides of the triangle. We find that D = p c = Da, which shows that the evians D and D a are isogonal ( p is the semi-perimeter of triangle, p = a+ b+ c). Theorem 9 D a

13 The triangle and its cotangent triangle are isogonal. We ll use theorem 8 and taking into account lemma 3, and the fact that the contact triangle and the triangle are homological, the homological center being the Gergonne s point. Observation 0 The homological center of the triangle and its cotangent triangle is called Nagel s point (N). Observation The Gergonne s point ( Γ ) and Nagel s point (N) are isogonal conjugated points. Theorem 0 The perpendiculars constructed on the sides of a triangle in the vertexes of the cotangent triangle are concurrent. The proof of this theorem results immediately using lema (arnot) Definition The concurrence point of the perpendiculars constructed in the vertexes of the cotangent triangle on the sides of the given triangle is called the evan s point ( V ). We will prove now that the reciprocal of the theorem of the orthohomological triangles is false We consider in a given triangle its contact triangle and also its cotangent triangle. The contact triangle and the triangle are homological, the homology center being the Geronne s point ( Γ ). The given triangle and its cotangent triangle are homological, their homological center being Nagel s point (N). even s point and the center of the inscribed circle have as podaire triangles the cotangent triangles and of contact, but these points are not isogonal conjugated (the point I is its own isogonal conjugate). References:.. Mihalescu, Geometria elementelor remarcabile, Ed. Tehnică, ucureşti, R.. Johnson, Modern Geometry: n Elementary Treatise on the Geometry of the Triangle and the ircle, 99. 3