Prove that a disk D that equals the sum of disks D1 of radius r1 and D2 of radiu

Question

Prove that a disk D that equals the sum of disks D1 of radius r1 and D2 of radius r2 necessarily has radius r satisfying r^2=r1^2+r2^2.

This is easy to do using the formula A=pi*r^2 but please do NOT use it!!

Try to do it using only the fact that if P1 and P2 are similar polygons with a pair of corresponding sides equal to r1 and r2 respectively, then P1:P2::D1:D2.

Use the results of Proposition 31 of Book 6 of the Elements, which asserts that the Pythagorean theorem holds for any similar polygon attached to the sides of a right triangle.

Explanation / Answer

area of disk D =area of disk D1 + area of disk D2

pi*r^2=pi*r1^2+pi*r2^2

r^2=r1^2+r2^2

hence proved

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