HMMT 二月 2020 · 冲刺赛 · 第 26 题

HMMT February 2020 — Guts Round — Problem 26

[15] Let ABCD be a cyclic quadrilateral, and let segments AC and BD intersect at E . Let W and Y be the feet of the altitudes from E to sides DA and BC , respectively, and let X and Z be the midpoints of sides AB and CD , respectively. Given that the area of AED is 9, the area of BEC is 25, and ◦ ∠ EBC − ∠ ECB = 30 , then compute the area of W XY Z .

解析

[15] Let ABCD be a cyclic quadrilateral, and let segments AC and BD intersect at E . Let W and Y be the feet of the altitudes from E to sides DA and BC , respectively, and let X and Z be the midpoints of sides AB and CD , respectively. Given that the area of AED is 9, the area of BEC is 25, ◦ and ∠ EBC − ∠ ECB = 30 , then compute the area of W XY Z . Proposed by: James Lin √ 15 Answer: 17 + 3 2 Solution: Reflect E across DA to E , and across BC to E . As ABCD is cyclic, 4 AED and W Y 4 BEC are similar. Thus E AED and EBE C are similar too. W Y Now since W is the midpoint of E E , X is the midpoint of AB , Y is the midpoint of EE , and Z is W Y the midpoint of DC , we have that W XY Z is similar to E AED and EBE C . W Y B X A E W W E Y Y E D Z C ◦ From the given conditions, we have EW : EY = 3 : 5 and ∠ W EY = 150 . Suppose EW = 3 x and √ √ EY = 5 x . Then by the law of cosines, we have W Y = 34 + 15 3 x . √ √ Thus, E E : W Y = 6 : 34 + 15 3. So by the similarity ratio, W ( √ ) ( ) √ 2 √ √ 34 + 15 3 34 + 15 3 15 [ W XY Z ] = [ E AED ] = 2 · 9 · = 17 + 3 . W 6 36 2