Hello, kicma!

On two sides of right angle we choose points A and B, |OA|=|OB|.

On arc AB with center in O we choose point P.

Through that point we draw line parallel with AB

which intersects OA in point C, and OB in point D.

Prove that: |PC|^{2}+|PD|^{2}=|AB|^{2.}

We have quarter circleCode:| D * | * | * B * * | * * P | * o | * . * r | . * * | . * * | . θ 45^{o}* 45^{o}* - + - - - - - - * - - -* - - - |O r A C

. .

Note that:

is on arc .

Let

We will use the Law of Sines.

. . . . .

. . . . .

Therefore: .