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# ​Let ABC be a triangle with\angle BAC = 90. A circle is tangent to the sides AB and AC at X and Y respectively, such that the points

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Let ABC be a triangle with\angle BAC = 90. A circle is tangent to the sides AB and AC at X and Y respectively, such that the points on the circle diametrically opposite X and Y both lie on the side BC. Given that AB = 6, find the area of the portion of the circle that lies outside the triangle.

Aug 8, 2020

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Let the center of a circle be O and the midpoint of BC be M

Angle B = 45º

AM = sin( 45º) * 6

XY = 2/3 * AM            (Segment XY is a side of a square inscribed in a circle.)

Area of an inscribed square    As = XY2

r = sin(45º) * XY

Area of a circle     Ac = r2pi

Area of a shaded segment    A = (Ac - As) / 4 = pi - 2

Aug 8, 2020
edited by Dragan  Aug 8, 2020
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Awesome solution Dragan!!

Guest Aug 8, 2020
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Find the area of the portion of the circle that lies outside the triangle.

Aug 8, 2020
edited by Omi67  Aug 8, 2020
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Amazing solution Omi!!!!!

Guest Aug 8, 2020