two circles in an equilateral triangle solution in Notes

# Solution to the [[Two Circles in an Equilateral Triangle]] Puzzle

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[[TwoCirclesinanEquilateralTriangle.png:pic]]

> What’s the side length of this equilateral triangle? Both circles have area $3$.
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## Solution by Lengths in an [[Equilateral Triangle]]

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[[TwoCirclesinanEquilateralTriangleAnnotated.png:pic]]
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With the points labelled as above, triangle $A E B$ is half of an [[equilateral triangle]] so the length of $A E$ is twice that of $E B$. This means that the length of $A D$ is five times the radius of one of the circles, which is $\sqrt{3}$. The height of an equilateral triangle is $\frac{\sqrt{3}}{2}$ times the length of one of its sides, so the side of the outer equilateral triangle is:

$$
\frac{2}{\sqrt{3}} \times 5 \sqrt{3} = 10
$$