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lines intersect on the inscribed sphere on a tetrahedron, concurrency

Source: Sharygin 2005 finals 11.6

August 31, 2019

geometry3D geometrytetrahedronsphereconcurrencyconcurrent

Problem Statement

The sphere inscribed in the tetrahedron

ABCD

touches its faces at points

A',B',C',D'

. The segments

AA'

and

BB'

intersect, and the point of their intersection lies on the inscribed sphere. Prove that the segments

CC'

and

DD'

also intersect on the inscribed sphere.