HMMT 二月 2024 · 几何 · 第 6 题

HMMT February 2024 — Geometry — Problem 6

In triangle ABC , a circle ω with center O passes through B and C and intersects segments AB and AC ′ ′ ′ ′ again at B and C , respectively. Suppose that the circles with diameters BB and CC are externally tangent to each other at T . If AB = 18 , AC = 36 , and AT = 12 , compute AO .

解析

In triangle ABC , a circle ω with center O passes through B and C and intersects segments AB and AC ′ ′ ′ ′ again at B and C , respectively. Suppose that the circles with diameters BB and CC are externally tangent to each other at T . If AB = 18 , AC = 36 , and AT = 12 , compute AO . Proposed by: Ethan Liu 65 Answer: 3 Solution 1: A ′ C ′ B M C M B T O B C By Radical Axis Theorem, we know that AT is tangent to both circles. Moreove, consider power ′ 2 ′ of a point A with respect to these three circles, we have AB · AB = AT = AC · AC . Thus 2 2 ′ 12 ′ 12 ′ ′ AB = = 8 , and AC = = 4 . Consider the midpoints M , M of segments BB , CC , B C 18 36 ◦ respectively. We have ∠ OM A = ∠ OM A = 90 , so O is the antipode of A in ( AM M ) . Notice B C B C AM AO B that △ AM T ∼ △ AOM , so = . Now, we can do the computations as follow: B C AM AT C AM · AM B C AO = AT ( ) ( ) ′ ′ AB + AB AC + AC 1 = 2 2 AT ( ) ( ) 8 + 18 36 + 4 1 65 = = . 2 2 12 3