HMMT 十一月 2019 · 冲刺赛 · 第 19 题

HMMT November 2019 — Guts Round — Problem 19

[11] Let ABC be a triangle with AB = 5, BC = 8, CA = 11. The incircle ω and A -excircle Γ are centered at I and I , respectively, and are tangent to BC at D and D , respectively. Find the ratio of 1 2 1 2 the area of 4 AI D to the area of 4 AI D . 1 1 2 2

解析

[11] Let ABC be a triangle with AB = 5, BC = 8, CA = 11. The incircle ω and A -excircle Γ are centered at I and I , respectively, and are tangent to BC at D and D , respectively. Find the ratio 1 2 1 2 of the area of 4 AI D to the area of 4 AI D . 1 1 2 2 Proposed by: Carl Schildkraut 1 Answer: 9 ′ ′ Let D and D be the points diametrically opposite D and D on the incircle and A -excircle, respec- 1 2 1 2 ′ tively. As I is the midpoint of D and D , we have x x x ′ [ AI D ] [ AD D ] 1 1 1 1 = . ′ [ AI D ] [ AD D ] 2 2 2 2 ′ ′ r s − a Now, 4 AD D and 4 AD D are homothetic with ratio = , where r is the inradius, r is the 1 2 A 1 2 r s A A -exradius, and s is the semiperimeter. Our answer is thus ( ) ( ) 2 s − a 4 1 = = . s 12 9