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Two half circles and a full circle fit inside a larger quarter circle as shown in the diagram. The centers of the two half circles are on the two sides of the quarter circle, respectively.

Prove that the triangle formed by the centers of three smaller circles, $\triangle O_1O_2O_3$, is a right triangle.

I was able to apply the Pythagorean formula to a few triangles involving the radii of the inscribed circles and conclude that the ratios of the three radii are 1:2:3. Then, the centers the three circles form a triangle with side ratios 3:4:5, hence, a right triangle.

On the other hand, I feel the proof may be an overkill, and evaluating the three radii explicitly may be unnecessary. There ought to be clean geometric solutions to show directly that the vertex $O_3$ is of a right angle, which I am not sure how to figure out.

Two half circles and a full circle fit inside a larger quarter circle as shown in the diagram. The centers of the two half circles are on the two sides of the quarter circle, respectively.

Prove that the triangle formed by the centers of three smaller circles, $\triangle O_1O_2O_3$, is a right triangle.

I was able to apply the Pythagorean formula to a few triangles involving the radii of the inscribed circles and proved that the ratios of the three radii are 1:2:3. Then, the centers of the three circles form a triangle with side-length ratios 3:4:5, hence, a right triangle.

On the other hand, I feel the proof may be an overkill, and evaluating the three radii explicitly may be unnecessary. There ought to be clean geometric solutions to show directly that the vertex $O_3$ is of a right angle, which I am not sure how to figure out.