HMMT 十一月 2011 · THM 赛 · 第 7 题
HMMT November 2011 — THM Round — Problem 7
题目详情
英文原题
- [ 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
英文解析
- [ 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 Athe 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 .
Y B A D ZIX − − →