A semicircle with an inscribed triangle. Both circles are tangent to the semicircle, the diameter and one other triangle side. Prove that the line segment connecting the two shown tangency points is parallel to the diameter.

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## Solution

This problem can be generalised to all inscribed triangles, not just right ones. I thank Ahmet ÇETİN for pointing this out. As shown in the diagram below, one can easily proof that the four tangency points form a parallelogram by using Sawayama’s Lemma.

## Poem

Out of focus

” I saw a glittering moon

I saw a beautifull morning

Shining on the hill

And I grabbed the cat by the tail

The future with my finger nails

But now I live in fear

It’s all out of focus

It’s all baby so unclear…..”

(Out of focus, Mick Jagger)