HMMT 二月 2005 · GEN2 赛 · 第 7 题
HMMT February 2005 — GEN2 Round — Problem 7
题目详情
- Three distinct lines are drawn in the plane. Suppose there exist exactly n circles in the plane tangent to all three lines. Find all possible values of n .
解析
- Three distinct lines are drawn in the plane. Suppose there exist exactly n circles in the 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 the three excircles. If the three lines concur or are all parallel, then there are 0 circles. If two lines are parallel and the third is not, then there are 2 circles lying between the two parallel lines, one on each side of the transverse line. These are the only possible configurations, so the answers are 0, 2, and 4. 2