HMMT 二月 2005 · 几何 · 第 9 题

英文解析

9. Let AC be a diameter of a circle ω of radius 1, and let D be the point on AC suchthat CD = 1 / 5. Let B be the point on ω such that DB is perpendicular to AC , andlet E be the midpoint of DB . The line tangent to ω at B intersects line CE at the point X . Compute AX .
   Solution: 3
   We first show that AX is perpendicular to AC . Let the tangent to ω at A intersect
   ′ ′ ′
   CB at Z and CE at X . Since ZA is parallel to BD and BE = ED , ZX = X A .
   ′
   Therefore, X is the midpoint of the hypotenuse of the right triangle ABZ , so it is also
   ′ ′ ′
   its circumcenter. Thus X A = X B , and since X A is tangent to ω and B lies on ω ,
   ′ ′
   we must have that X B is tangent to ω , so X = X .
   4 3 3
   Let O be the center of ω . Then OD = , so BD = and DE = . Then AX =
   5 5 10
   AC 3 2
   DE · = · = 3.
   DC 10 1 / 5
   X’Z
   CAEB
   3DO