HMMT 十一月 2011 · THM 赛 · 第 8 题

HMMT November 2011 — THM Round — Problem 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 .

解析

[ 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 2 16 = P D = ( P F )( P E ) = 2( P E ) ⇒ P E = 8 . ◦ 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 tangents the tangent at D must be perpendicular to the radius containing D . Hence, the area of the circle is 1 2 ( DE ) π = 12 π . 2 D 4 P 2 √ F 2 3 6 E