MathDB

P24

Part of 2015 Sharygin Geometry Olympiad

Problems(1)

4 lines concurrent, in an interior point, starting with insphere of tetrahedron

Source: 2015 Sharygin Geometry Olympiad Correspondence Round 24

8/2/2018
The insphere of a tetrahedron ABCD with center OO touches its faces at points A1,B1,C1A_1,B_1,C_1 and D1D_1. a) Let PaP_a be a point such that its reflections in lines OB,OCOB,OC and ODOD lie on plane BCDBCD. Points Pb,PcP_b, P_c and PdP_d are defined similarly. Prove that lines A1Pa,B1Pb,C1PcA_1P_a,B_1P_b,C_1P_c and D1PdD_1P_d concur at some point P P. b) Let II be the incenter of A1B1C1D1A_1B_1C_1D_1 and A2A_2 the common point of line A1IA_1I with plane B1C1D1B_1C_1D_1. Points B2,C2,D2B_2, C_2, D_2 are defined similarly. Prove that PP lies inside A2B2C2D2A_2B_2C_2D_2.
spheretetrahedron3-Dimensional Geometrygeometry3D geometry