The tipi entrance

A semicircle with three tangents. Prove that purple : green = red : yellow.

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After some angle chasing, one finds that triangles KIF and JKH are similar. This gives p/x=x/(r+y), so p(r+y)=x2. From similarity of triangles FKG and HEK we get (p+g)/x=x/r, so r(p+g)=x2. Combining these two equations gives py=rg.

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