HMMT 十一月 2011 · THM 赛 · 第 7 题

HMMT November 2011 — THM Round — Problem 7

[ 4 ] Let XY Z be a triangle with ∠ XY Z = 40 and ∠ Y ZX = 60 . A circle Γ, centered at the point I , lies inside triangle XY Z and is tangent to all three sides of the triangle. Let A be the point of − → tangency of Γ with Y Z , and let ray XI intersect side Y Z at B . Determine the measure of ∠ AIB . − − →

解析

[ 4 ] Let XY Z be a triangle with ∠ XY Z = 40 and ∠ Y ZX = 60 . A circle Γ, centered at the point I , lies inside triangle XY Z and is tangent to all three sides of the triangle. Let A be the point of − → tangency of Γ with Y Z , and let ray XI intersect side Y Z at B . Determine the measure of ∠ AIB . ◦ Answer: 10 Let D be the foot of the perpendicular from X to Y Z . Since I is the incenter and A the point of tangency, IA ⊥ Y Z , so AI ‖ XD ⇒ ∠ AIB = ∠ DXB. Since I is the incenter, 1 1 ◦ ◦ ◦ ◦ ∠ BXZ = ∠ Y XZ = (180 − 40 − 60 ) = 40 . 2 2 Consequently, we get that ◦ ◦ ◦ ◦ ∠ AIB = ∠ DXB = ∠ ZXB − ∠ ZXD = 40 − (90 − 60 ) = 10 . X I Y B A D Z − − →