Beautiful Geometry 1

Two Arcs in a Square

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

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.

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