HMMT 二月 2026 · 冲刺赛 · 第 9 题

HMMT February 2026 — Guts Round — Problem 9

[7] Let ABCD be a rectangle. Let E be the reflection of C over B . The circumcircle of triangle ACE intersects line CD at a point F ̸ = C . Given that AC = 8 and AF = 6 , compute the area of rectangle ABCD .

解析

[7] Let ABCD be a rectangle. Let E be the reflection of C over B . The circumcircle of triangle ACE intersects line CD at a point F ̸ = C . Given that AC = 8 and AF = 6 , compute the area of rectangle ABCD . Proposed by: Rishabh Das 768 Answer: = 30 . 72 25 Solution: ©2026 HMMT E A B D F C Since B is the midpoint of CE , and AB ⊥ CE , the circumcenter of triangle ACE lies on line AB . As AB ⊥ AD , line AD is tangent to the circumcircle of triangle ACE . Thus, ∠ F AD = ∠ ACD , and so △ F AD ∼ △ ACD . Therefore, 3 AF AD = = . 4 CA CD The above implies that AD = 3 x and CD = 4 x for some x > 0 . Now, applying the Pythagorean theorem on triangle ACD yields 2 2 2 2 2 2 64 = AC = AD + CD = (3 x ) + (4 x ) = 25 x . Finally, we compute that the area of ABCD is 64 768 AD · CD = (3 x ) · (4 x ) = 12 · = . 25 25