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On the side AB of the square ABCD, an equilateral triangle AKB was constructed (outside). Find the radius of the circle circumscribed around triangle CKD, if $\mathrm{AB}=1$.

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Construct an equilateral triangle on side $\mathrm{CD}$ inside the square $\mathrm{ABCD}$.

## Solution

Construct an equilateral triangle CDM on side CD inside the square ABCD. Then CM=1 and DM=1. Moreover, triangle ABK is obtained from triangle CDM by a parallel translation by vector CB, the length of which is 1. Therefore, KM equals 1. Thus, point M is at a distance of 1 from points C, D, and K. This means that M is the center of the circumcircle of triangle CDK, and the radius of this circle is 1.

## Problem