PUMaC 2014 · 几何（B 组） · 第 3 题

PUMaC 2014 — Geometry (Division B) — Problem 3

[ 4 ] As given in figure (not drawn to proportion), in 4 ABC , E ∈ AC , D ∈ AB , P = BE ∩ CD Given that S 4 BP C = 12, while the areas of 4 BP D , 4 CP E and quadrilateral AEP D are all the same, which is x . Find the value of x .

解析

[ 4 ] As given in figure (not drawn to proportion), in 4 ABC , E 2 AC , D 2 AB , P = BE \ CD Given that S 4 BP C = 12, while the areas of 4 BP D , 4 CP E and quadrilateral AEP D are all the same, which is x . Find the value of x . Solution: z 4 AEF 4 AEB Let S 4 ADF = y and S 4 AEF = z . Hence x = y + z . We see that = = = x 4 CEF 4 CEB 2 x y 4 ADF 4 ADC 2 x 4 x and = = = . Solving, we see that 1 = and hence x + 12 x 4 BDF 4 BDC x + 12 x + 12 x = 4 .