HMMT 二月 2005 · GEN2 赛 · 第 7 题

英文解析

7. Three distinct lines are drawn in the plane. Suppose there exist exactly n circles inthe plane tangent to all three lines. Find all possible values of n .
   Solution: 0 , 2 , 4
   If the three lines form a triangle, then there are 4 circles, namely the incircle and thethree excircles. If the three lines concur or are all parallel, then there are 0 circles. Iftwo lines are parallel and the third is not, then there are 2 circles lying between thetwo parallel lines, one on each side of the transverse line. These are the only possibleconfigurations, so the answers are 0, 2, and 4. 2