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
![](https://mirangu.com/wp-content/uploads/2022/05/AsWithinSoWithoutSolution.jpg)
Note that this solution never uses the right angle at P. The prove therefore is also valid for non-diameter chords.
![](https://mirangu.com/wp-content/uploads/2022/05/AsWithinSoWithoutGeneralised.jpg)
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
![](https://mirangu.com/wp-content/uploads/2022/05/AsWithinSoWithoutBella.jpg)
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.