HMMT 十一月 2011 · THM 赛 · 第 6 题

英文解析

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 onsegment 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 = .
   Let O , which must also lie on this line, be the center of ω , and let D be the point of tangency between 2
   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 = .
   A2
   B Q CPOD
   ° °