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
