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projection of intersection of external <C-bisector and AB, on OI

Source: 2009 Sharygin Geometry Olympiad Final Round problem 3 grade 10

July 26, 2018
geometrydistanceangle bisector

Problem Statement

The cirumradius and the inradius of triangle ABCABC are equal to RR and r,O,Ir, O, I are the centers of respective circles. External bisector of angle CC intersect ABAB in point PP. Point QQ is the projection of PP to line OIOI. Find distance OQ.OQ.
(A.Zaslavsky, A.Akopjan)