An equilateral triangle containing two circles of given area and a common tangent divided in two. What is the proportion of the two parts?

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## Solution

The proportion a : b is 2 : 1.

By using the tangent-tangent theorem multiple times, we can label the segments as shown, using also that the right radius is twice the left one. Now since the left side equals the right side, we find that a+x+r√3=a+2r√3 and hence x=r√3. Since also the bottom side equals the other sides, we get that a=2b.

## Poem

The single parenting is a breeze

Going on holiday is not easy

Be a good organiser firstly

And over all be very patient, very

Discuss everything together gently

Don’t forget the mobile phone, essentially

And be very zen eventually

That’s the recipe for a good holiday eventually