HMMT 二月 2008 · 几何 · 第 3 题

英文解析

3. [ 4 ] Let ABC be a triangle with ∠ BAC = 90 . A circle is tangent to the sides AB and AC at X and
   Y respectively, such that the points on the circle diametrically opposite X and Y both lie on the side
   BC . Given that AB = 6, find the area of the portion of the circle that lies outside the triangle.
   X BAYC
   ′ ′
   Answer: π − 2 Let O be the center of the circle, and r its radius, and let X and Y be the points
   ′ ′ ′ ′ °
   diametrically opposite X and Y , respectively. We have OX = OY = r , and ∠ X OY = 90 . Since
   ′ ′ ′′ ′
   triangles X OY and BAC are similar, we see that AB = AC . Let X be the projection of Y onto
   ′′ ′ ′′ ′ ′′
   AB . Since X BY is similar to ABC , and X Y = r , we have X B = r . It follows that AB = 3 r , sor = 2.
   ′C
   ′X
   O Y
   ′′Y
   X BA
   1X
   ′ ′
   Then, the desired area is the area of the quarter circle minus that of the triangle X OY . And the
   1 1
   2 2
   answer is πr − r = π − 2.
   4 2