HMMT 十一月 2011 · THM 赛 · 第 6 题

HMMT November 2011 — THM Round — Problem 6

[ 3 ] Let ABC be an equilateral triangle with AB = 3. Circle ω with diameter 1 is drawn inside the triangle such that it is tangent to sides AB and AC . Let P be a point on ω and Q be a point on segment BC . Find the minimum possible length of the segment P Q . ◦ ◦

解析

[ 3 ] Let ABC be an equilateral triangle with AB = 3. Circle ω with diameter 1 is drawn inside the triangle such that it is tangent to sides AB and AC . Let P be a point on ω and Q be a point on segment BC . Find the minimum possible length of the segment P Q . Theme Round √ 3 3 − 3 Answer: Let P , Q , be the points which minimize the distance. We see that we want both 2 √ 3 3 to lie on the altitude from A to BC . Hence, Q is the foot of the altitude from A to BC and AQ = . 2 Let O , which must also lie on this line, be the center of ω , and let D be the point of tangency between 1 1 ◦ ω and AC . Then, since OD = , we have AO = 2 OD = 1 because ∠ OAD = 30 , and OP = . 2 2 Consequently, √ 3 3 − 3 P Q = AQ − AO − OP = . 2 A D O P B Q C ◦ ◦