HMMT 十一月 2011 · THM 赛 · 第 8 题

英文解析

8. [ 5 ] Points D, E, F lie on circle O such that the line tangent to O at D intersects ray EF at P . Given
   °
   that P D = 4, P F = 2, and ∠ F P D = 60 , determine the area of circle O .
   Theme Round
   Answer: 12 π By the power of a point on P , we get that
   16 = P D = ( P F )( P E ) = 2( P E ) ⇒ P E = 8 .2
   °
   However, since P E = 2 P D and ∠ F P D = 60 , we notice that P DE is a 30 − 60 − 90 triangle, so
   √
   DE = 4 3 and we have ED ⊥ DP . It follows that DE is a diameter of the circle, since tangentsthe tangent at D must be perpendicular to the radius containing D . Hence, the area of the circle is
   ( DE ) π = 12 π .21
   D 4 P2
   √2
   2 3F
   E6