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DL^2 + KC^2 = AB^2 , semicircle externally on a rectangle

Source: Polish MO Recond Round 1979 p6

September 9, 2024

geometrysemicirclerectangle

Problem Statement

On the side

\overline{DC}

of the rectangle

ABCD

in which

\frac{AB}{AD} = \sqrt{2}

a semicircle is built externally. Any point

M

of the semicircle is connected by segments with

A

and

B

to obtain points

K

and

L

\overline{DC}

, respectively. Prove that

DL^2 + KC^2 = AB^2