HMMT 十一月 2024 · GEN 赛 · 第 7 题

HMMT November 2024 — GEN Round — Problem 7

Let triangle ABC have AB = 5 , BC = 8 , and ∠ ABC = 60 . A circle ω tangent to segments AB and BC intersects segment CA at points X and Y such that points C , Y , X , and A lie along CA in this order. If ω is tangent to AB at point Z and ZY ∥ BC , compute the radius of ω .

解析

Let triangle ABC have AB = 5 , BC = 8, and ∠ ABC = 60 . A circle ω tangent to segments AB and BC intersects segment CA at points X and Y such that points C , Y , X , and A lie along CA in this order. If ω is tangent to AB at point Z and ZY ∥ BC , compute the radius of ω . Proposed by: Ethan Liu √ 40 3 40 √ Answer: = 39 13 3 Solution: A X Z Y ◦ 60 B C T ◦ Let ω tangent to BC at T . Observe that BT = BZ and ∠ ABC = 60 , so △ T BZ is equilateral. Moreover, the tangent to ω at T is parallel to BC , so T Y = T Z . Combining this with ∠ T ZY = ◦ ∠ ZT B = 60 , it follows that △ T Y Z is equilateral as well. Now, let BT = BZ = T Z = T Y = Y Z = x . Then, AZ = 5 − x . Thus, similar triangles AY Z and ABC gives AZ Y Z 5 − x x = = ⇒ = . AB BC 5 8 40 Solving this equation gives 40 − 8 x = 5 x , or x = . Finally, since △ T Y Z is an equilateral triangle 13 x 40 √ √ inscribed in ω , the radius of ω is = . 3 13 3