MathDB

M2688

Part of Kvant 2022

Problems(1)

Collinear points and incircle

Source: Kvant Magazine No. 2 2022 M2688

3/8/2023
Let Ta,TbT_a, T_b and TcT_c be the tangent points of the incircle Ω\Omega of the triangle ABCABC with the sides BC,CABC, CA and ABAB{} respectively. Let X,YX, Y and ZZ{} be points on the circle Ω\Omega such that AA{} lies on the ray YXYX, BB{} lies on the ray ZYZY and CC{} lies on the ray XZXZ. Let PP{} be the intersection point of the segments ZXZX and TbTcT_bT_c, and similarly Q=XY∩TcTaQ=XY \cap T_cT_a and R=YZ∩TaTbR=YZ\cap T_aT_b. Prove that the points A,BA, B and CC{} lie on the lines RP,PQRP, PQ and QRQR{}, respectively.
Proposed by L. Shatunov (11th grade student)
Kvantgeometry