HMMT 二月 1998 · 几何 · 第 9 题
HMMT February 1998 — Geometry — Problem 9
题目详情
英文原题
Question Nine . [7 points]
Let T be the intersection of the common internal tangents of circles
C , C with centers O , O respectively. Let P be one of the points
1 2 1 2
of tangency on C and let line l bisect angle O TP . Label the
1 1
intersection of l with C that is farthest from T , R , and label theintersection of l with C that is closest to T , S . If C has radius 4,1
2 1
C has radius 6, and O O = 20 , calculate ( TR )( TS ).
2 1 2
2C
S OTPR
2O
C1
l 1
解析
英文解析
- The dilation of ratio − about T sends C to C , O to O , and S to the other intersection
2 1 2 1
3 33
of s with C , which we shall call U . We can now compute T R · T S = T R · T U = T P =2
2 21
( ) ( )
( ) ( )
2 2
O O
3 2 2 3 2 3 20 2
1 2
( O T − O P ) = − O P = − 4 = 72 .
1 1 1
2 2 1+3 / 2 2 5 / 2