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Three pairwise tangent cicrcles

Source: Iranian National Olympiad (3rd Round) 2007

August 27, 2007
geometrygeometric transformationrotationradical axisgeometry proposed

Problem Statement

Let ABC ABC be a triangle, and D D be a point where incircle touches side BC BC. M M is midpoint of BC BC, and K K is a point on BC BC such that AK⊥BC AK\perp BC. Let D′ D' be a point on BC BC such that D′MD′K=DMDK \frac{D'M}{D'K}=\frac{DM}{DK}. Define ωa \omega_{a} to be circle with diameter DD′ DD'. We define ωB,ωC \omega_{B},\omega_{C} similarly. Prove that every two of these circles are tangent.
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