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4

Part of 2023 Oral Moscow Geometry Olympiad

Problems(2)

The center of (IPQ) is on BC

Source: Oral Moscow geometry olympiad 2023 8-9.4

4/20/2023
Let II be the incenter of triangle ABCABC, tangent to sides ABAB and ACAC at points EE and FF, respectively. The lines through EE and FF parallel to AIAI intersect lines BIBI and CICI at points PP and QQ, respectively. Prove that the center of the circumcircle of triangle IPQIPQ lies on line BCBC.
geometry
concurrency in 3D, lines connecting tyoucpoints of incircle and excircle

Source: Oral Moscow Geometry Olympiad 2023 10-11 p4

2/26/2024
Given isosceles tetrahedron PABCPABC (faces are equal triangles). Let A0A_0, B0B_0 and C0C_0 be the touchpoints of the circle inscribed in the triangle ABCABC with sides BCBC, ACAC and ABAB respectively, A1A_1, B1B_1 and C1C_1 are the touchpoints of the excircles of triangles PCAPCA, PABPAB and PBCPBC with extensions of sides PAPA, PBPB and PCPC, respectively (beyond points AA, BB, CC). Prove that the lines A0A1A_0A_1, B0B1B_0B_1 and C0C1C_0C_1 intersect at one point.
geometryconcurrency3D geometry