HMMT 二月 2020 · 几何 · 第 4 题

HMMT February 2020 — Geometry — Problem 4

Let ABCD be a rectangle and E be a point on segment AD . We are given that quadrilateral BCDE has an inscribed circle ω that is tangent to BE at T . If the incircle ω of ABE is also tangent to BE 1 2 at T , then find the ratio of the radius of ω to the radius of ω . 1 2

解析

Let ABCD be a rectangle and E be a point on segment AD . We are given that quadrilateral BCDE has an inscribed circle ω that is tangent to BE at T . If the incircle ω of ABE is also tangent to BE 1 2 at T , then find the ratio of the radius of ω to the radius of ω . 1 2 Proposed by: James Lin √ 3+ 5 Answer: 2 Solution: Let ω be tangent to AD , BC at R , S and ω be tangent to AD , AB at X , Y . Let 1 2 AX = AY = r , EX = ET = ER = a , BY = BT = BS = b . Then noting that RS ‖ CD , we see that ABSR is a rectangle, so r + 2 a = b . Therefore AE = a + r , AB = b + r = 2( a + r ), and so √ √ 1+ 5 BE = ( a + r ) 5. On the other hand, BE = b + a = r + 3 a . This implies that a = r . The desired 2 √ RS AB a + r 3+ 5 ratio is then = = = . 2 AY 2 r r 2 A Y B X T E R S D C