SOME NEW THEOREMS IN PLANE GEOMETRY II

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1 SOME NEW THEOREMS IN PLANE GEOMETRY II ALEXANDER SKUTIN 1. Introduction This work is an extension of [1]. In fact, I used the same ideas and sections as in [1], but introduced other examples of applications. 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 Fermat points. Theorem 2.1. Consider triangle ABC with the first Fermat point F. Let A B C be the cevian triangle of F wrt ABC. Let F A be the second Fermat point of B C F. Similarly define F B, F C. Let M A be the midpoint of AF. Similarly define M B and M C. Then the triangles F A F B F C and M A M B M C are perspective. Theorem 2.2. Consider triangle ABC with the first Fermat point F. Let F A be the second Fermat point for BCF. Similarly define F B, F C. Let FA be the second Fermat point for F B F C F. Similarly define FB, F C. Then (1) Lines F A F A, F BF B, F CF C are concurrent at the first Fermat point of F AF B F C. (2) The second Fermat point of F A F B F C lies on (F A F B F C ). Theorem 2.3. Consider triangle ABC with the first Fermat point F 1 and the second Fermat point F 2, let A B C be the pedal triangle of F 1 wrt ABC. Consider isogonal conjugations A B, A C of A wrt AF 1 B, AF 1 C respectively. Let F A be the second Fermat point of F 1 A B A C. Similarly define F B, F C. Then the points F 2, F A, F B, F C lie on the same circle Incircles. Theorem 2.4. Consider triangle ABC, let A excircle is tangent to sides of ABC at A A B A C A. Similarly define the points A B, B B, C B, A C, B C, C C. Let A B B B meet A C C C at A. Similarly define B, C. Let A be reflection of A wrt A. Similarly define B, C. Then I is the incenter of A B C Mathematics Subject Classification. 51M04, 51N20. Key words and phrases. Plane geometry, Analytic geometry. 1

2 2 ALEXANDER SKUTIN 2.4. Nine-point circles. Theorem 2.5. Consider triangle ABC with the nine - point center N and centroid G. Let N A be the nine - point center of BCG. Similarly define N B, N C. Then N is centroid of N A N B N C. And G lies on the nine - point circle of ABC Tangent circles. Theorem 2.6. Consider triangle ABC with orthocenter H, let circle ω A is tangent to AB, AC and externally tangent to (BHC) at A. Similarly define B, C and ω B, ω C. Then (1) Points A, B, C, H lie on the same circle. (2) Let circle Ω is tangent to ω A, ω B, ω C. Then H lies on Ω Isogonal conjugations. Theorem 2.7. Consider triangle ABC with the first Fermat point F and let A B C be the cevian triangle of F wrt ABC. Let F A be isogonal to F wrt AB C and let F A be isogonal to F A wrt ABC. Similarly define F B, F C. Then the points F A, F B, F C form an equilateral triangle with center at F. Theorem 2.8. Consider triangle ABC with the first Fermat point F and let A B C be the cevian triangle of F wrt ABC. Let F A be isogonal to A wrt F B C and let F A be isogonal to F A wrt ABC. Similarly define F B, F C. Let F A be reflection of F A wrt BC. Similarly define FB, F C. Then F coincide with the first Fermat point of F A F B F C. Theorem 2.9. Consider triangle ABC with incenter I and incircle ω which is tangent to BC, CA, AB at A, B, C respectively. Let A B, A C be isogonal conjugations of A wrt AIB, AIC respectively. Similarly define B A, B C, C A, C B. Then the midpoints of A B A C, B A B C, C A C B lie on ω. Remark 2.1. In all theorems from this section consider the case of an equilateral triangle. 3. Construction of midpoint analog Definition 3.1. For any pairs of points A, B and C, D denote M(AB, CD) as the Miquel point of the complete quadrilateral formed by the four lines AC, AD, BC, BD. Definition 3.2. For any point X and a segment Y Z denote M(X, Y Z) as a point P, such that the circles (P XY ) and (P XZ) are tangent to segments XZ, XY at X. Consider any two segments AB and CD, then the point M(AB, CD) can be seen as midpoint between the two segments AB, CD. Also we can consider the segment AB and the point C and to look on the point M(C, AB) as on the midpoint between the point C and the segment AB. Remark 3.1. In the case when A = B and C = D we will get that the points M(C, AB) and M(AB, CD) are midpoints of AC. Theorem 3.1. Consider eight lines l j i, where 1 i 4, j = 1, 2. Let given that l1 i l2 i for any 1 i 4. Let M k be the Miquel point for l j 1, lj 2, lj 3, lj 4, j = 1, 2. By definition let P j pq = l j p l j q and let Q pq = M(M 1 M 2, P 1 pqp 2 pq). Then the circles (Q 12 Q 23 Q 31 ), (Q 12 Q 24 Q 41 ), (Q 13 Q 34 Q 41 ), (Q 23 Q 34 Q 42 ) passes through the same point.

3 SOME NEW THEOREMS IN PLANE GEOMETRY II 3 4. Combination of different facts Here we will build combinations of some well-known constructions from geometry. Theorem 4.1 (Pappus and Mixtilinear circles). Consider triangle ABC, let A mixtilinear incircle ω A is tangent to (ABC) at A 1 and A mixtilinear excircle Ω A is tangent to (ABC) at A 2. Similarly define B 1, B 2, C 1, C 2 and ω B, ω C, Ω B, Ω C. Let the radical line of ω A, ω B meet A 1 B 1 at X and the radical line of Ω A, Ω B meet A 2 B 2 at Y. Let l C be the line through the points A 1 Y A 2 X, A 1 B 2 A 2 B 1 and XB 2 Y B 1. Similarlty define l B, l C. Then the triangle formed by the lines l A, l B, l C is perspective to ABC. Theorem 4.2 (Conics and Japanese theorem). Consider cyclic quadrilateral ABCD. Let I A, I B, I C, I D be the incenters of DAC, ABC, BCD, CDA respectively. Consider any point P on (ABCD) and any conic C AC through ACI B I D P and a conic C BD through BDI A I C P. Then the conics C AC, C BD meet on (ABCD) at two different points Combinations with radical lines. Theorem 4.3 (Japanese theorem and radical lines). Consider cyclic quadrilateral ABCD. Let I A, I B, I C, I D be the incenters of DAC, ABC, BCD, CDA respectively. Then circles with diameters AI A, BI B, CI C, DI D have the same radical center. Theorem 4.4 (Morley s theorem and radical lines). Consider any triangle ABC and it s three external Morley s triangles A A B A C A, A B B B C B, A C B C C C (see picture below for more details). Let l A be the radical line of (A A A B A C ) and (C A B A B B C C ). Similarly define the lines l B, l C. Then the lines l A, l B, l C form a triangle which is perspective to ABC. A B A C A C B B B C C B C C B A A C A B A

4 4 ALEXANDER SKUTIN 5. Facts related to the set of confocal conics Definition 5.1. Consider 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 A, B. Consider the intersection point X of the tangents to the conic K 1 from A, B. Similarly let Y be the intersection point of the tangents to the conic K 2 from A, B. Then let by definition L F (K 1, K 2 ) = XY. Next fact can be seen also as a fact which is related to [1, Section 6]. Theorem 5.1. Consider triangle ABC with orthocenter H. Let H A be reflection of H wrt BC. Similarly define H B and H C. Let I AHB be the incenter of a triangle formed by lines AC, H B H C and AH B. And let C AHB be a conic with foci at A and H B and which goes through I AHB. Similarly define the conics C CHB, C AHC, C BHC, C CHA, C BHA. Let A be the second intersection point of L A (C AHB, C AHC ) with (ABC). Similarly define B, C. Then (1) Line L HA (C BHA, C CHA ) is tangent to (ABC) at H A. (2) Lines AA, BB, CC are concurrent. A H C H B H C B H A

5 SOME NEW THEOREMS IN PLANE GEOMETRY II 5 6. Fun with some conjugations In this section we will construct some nice theorems which includes some famous conjugations. Theorem 6.1. Consider triangle ABC. Let I be the incenter of ABC and G be the centroid of ABC. Let I be isotomic conjugation of I wrt ABC and G be isogonal conjugation of G wrt ABC. Let X = II GG and X 1, X 2 be isogonal and isotomic conjugations of X wrt ABC respectively. Then the points G, I, X 1, X 2 lie on the same line. Next, it will be useful to use the notation P G for the isotomic conjugation of the point P wrt ABC and P I for the isogonal conjugation of P wrt ABC. Theorem 6.2. Consider triangle ABC. Let I be the incenter of ABC and G be the centroid of ABC. Let X = II G GG I, Y = IG I GI G. Then (1) Point Y I lies on GG I. (2) Point Y G lies on II G. (3) Let the lines Y I X I and Y G X G meet at Z. Then Z I lies on IG I. (4) Point Z G lies on GI G. (5) Let T = IG I G G I. Then the points X, Y, T I, T G lie on the same line. References [1] A. Skutin, Some new theorems in plane geometry [2] solver6, Math blog