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Intermediate

As within, so without

A semicircle and a circle are tangent. The tangency point is shown. Prove that the two coloured triangles are similar.

Scroll down for a solution to this problem.

Solution

Solution by Kamran Zamani.

Note that this solution never uses the right angle at P. The prove therefore is also valid for non-diameter chords.

Generalised version

Poem

As within so without
With a triangle in interactions
And power and responsability in connection
In this triangulation
Each one has a singular position
People seem not able to take decisions
In Karpmann’s triangle, it’s difficult to analyse
And to know well the motivations

Bella

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One reply on “As within, so without”

Absolutely so.
At first I was going to make use of the diameter. But when I labeled alpha and beta, I realized that I didn’t have to after all. Thanks for bringing the generality of it to my attention as it had not occurred to me.

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