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For the triangle below, let x be the area of the circumcircle, and let y be the area of the incircle. Compute x - y.

 

 Oct 6, 2023
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Because △ABC is a right triangle, it follows that the circumcenter of △ABC is the midpoint of the hypotenuse. Thus we have OA=OB=OC=13. By Pythagoras, we also have AC2+BC2=AB2, so 21​(AB+BC)(AB−BC)=AB2, so [AB = \frac{2AC^2}{AB + BC} = \frac{2 \cdot 24^2}{24 + 10} = \boxed{36}.]

The radius of the incircle of △ABC is [r = \frac{K}{s} = \frac{2(24)(10)(26)}{(24 + 10 + 26)} = 2.]Then the area of the incircle is K=πr2=4π​, so x−y=362π−4π=36π(36−1)​=1246π​.

 Oct 6, 2023

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