Beautiful Geometry 1

Two Arcs in a Square

A semicircle and quarter circle inside a square. How long is the black tangent line?

SOLUTION

Connect the center $B$ of the quarter-circle and the center $E$ of the semi-circle. We have $BE = 5 + 8 = 13$ .

Since the black segment $FG$ is tangent to the two circles, it must be orthogonal to $BE$ . Since they connect the two pairs of opposite sides of the square, it is easy to see that they are equal.

One way to show this is by constructing a perpendicular $FJ$ from $F$ to $BC$ . We have that $FJ = AB = BC$ , $∠JGF = 180° - ∠FGC = ∠CEB$ , and $∠FJG = ∠BCE = 90°$ . Therefore, $△FJG \cong △BCE$ , and $FG = BE = 13$ .

Two Arcs in a Square

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