HMMT 二月 2021 · 几何 · 第 8 题

HMMT February 2021 — Geometry — Problem 8

Two circles with radii 71 and 100 are externally tangent. Compute the largest possible area of a right triangle whose vertices are each on at least one of the circles.

解析

Two circles with radii 71 and 100 are externally tangent. Compute the largest possible area of a right triangle whose vertices are each on at least one of the circles. Proposed by: David Vulakh Answer: 24200 Solution: B O C A In general, let the radii of the circles be r < R , and let O be the center of the larger circle. If both endpoints of the hypotenuse are on the same circle, the largest area occurs when the hypotenuse is a 2 diameter of the larger circle, with [ ABC ] = R . If the endpoints of the hypotenuse are on different circles (as in the diagram above), then the distance from O to AB is half the distance from C to AB . Thus [ ABC ] = 2[ AOB ] = AO · OB sin ∠ AOB. ◦ AO · OB and sin ∠ AOB are simultaneously maximized when AO · OB = (2 r + R ) · R and m ∠ AOB = 90 , 2 so the answer is R + 2 Rr = 24200.