HMMT 二月 2024 · 几何 · 第 8 题

HMMT February 2024 — Geometry — Problem 8

Let ABT CD be a convex pentagon with area 22 such that AB = CD and the circumcircles of triangles ◦ ◦ T AB and T CD are internally tangent. Given that ∠ AT D = 90 , ∠ BT C = 120 , BT = 4 , and CT = 5 , compute the area of triangle T AD . ◦

解析

Let ABT CD be a convex pentagon with area 22 such that AB = CD and the circumcircles of triangles ◦ ◦ T AB and T CD are internally tangent. Given that ∠ AT D = 90 , ∠ BT C = 120 , BT = 4 , and CT = 5 , compute the area of triangle T AD . Proposed by: Pitchayut Saengrungkongka √ Answer: 64(2 − 3) ∼ Solution: Paste △ T CD outside the pentagon to get △ ABX △ DCT . From the tangent circles = condition, we get ◦ ∠ XBT = 360 − ∠ XBA − ∠ ABT ◦ = 360 − ∠ DCT − ∠ ABT ◦ ◦ ◦ = 360 − 270 = 90 ◦ ∠ XAT = 90 − ∠ BXA − ∠ AT B ◦ = 90 − ∠ CT D − ∠ AT B ◦ ◦ ◦ ◦ = 90 − (120 − 90 ) = 60 . Moreover, if x = AT and y = T D , then notice that [ ABT CD ] = [ ABT ] + [ CDT ] + [ AT D ] = [ XAT ] − [ XBT ] + [ AT D ] 1 1 1 ◦ = xy sin 60 − · 4 · 5 + xy 2 2 2 √ 2 + 3 = xy − 10 , 4 so we have √ √ 4 1 xy = 32 · √ = 128(2 − 3) = ⇒ [ AT D ] = xy = 64(2 − 3) . 2 2 + 3 T C B D A X T 5 5 4 C y y x D B ◦ 60 A ◦