MathDB

19

Part of 2013 Sharygin Geometry Olympiad

Problems(1)

Coincidence of concurrency points

Source: Sharygin First Round 2013, Problem 19

4/8/2013
a) The incircle of a triangle ABCABC touches ACAC and ABAB at points B0B_0 and C0C_0 respectively. The bisectors of angles BB and CC meet the perpendicular bisector to the bisector ALAL in points QQ and PP respectively. Prove that the lines PC0,QB0PC_0, QB_0 and BCBC concur.
b) Let ALAL be the bisector of a triangle ABCABC. Points O1O_1 and O2O_2 are the circumcenters of triangles ABLABL and ACLACL respectively. Points B1B_1 and C1C_1 are the projections of CC and BB to the bisectors of angles BB and CC respectively. Prove that the lines O1C1,O2B1,O_1C_1, O_2B_1, and BCBC concur.
c) Prove that the two points obtained in pp. a) and b) coincide.
geometrycircumcircleanalytic geometrygeometric transformationhomothety