The smaller circle has its centre on the larger circle and is also the triangle’s incircle. Prove that the circle centres and the triangle vertex are collinear.

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## Solution

The side lengths of the triangle that are outside the big circle are the same length. Because there are 2 congruent right angled triangles there. For the single hatched lengths, use the Tangent-tangent theorem. For the double hatched lengths, use the fact that equal angles implies equal chords. And for the triple hatched lengths, use RHS congruency.

## Poem

In orbit

Moving in space

Floating in a beautiful universe

Piercing gravity

Moon and sun relied together

But in different circles

Sharing the night

The stars

The sky

Dreaming of a big blast

And then come back to earth

For a sunset

So near, so far

To our planet

For which we care