HMMT 十一月 2018 · GEN 赛 · 第 8 题

HMMT November 2018 — GEN Round — Problem 8

Equilateral triangle ABC has circumcircle Ω. Points D and E are chosen on minor arcs AB and AC of Ω respectively such that BC = DE . Given that triangle ABE has area 3 and triangle ACD has area 4, find the area of triangle ABC .

解析

Equilateral triangle ABC has circumcircle Ω. Points D and E are chosen on minor arcs AB and AC of Ω respectively such that BC = DE . Given that triangle ABE has area 3 and triangle ACD has area 4, find the area of triangle ABC . Proposed by: Yuan Yao 37 Answer: 7 ◦ A rotation by 120 about the center of the circle will take ABE to BCD , so BCD has area 3. Let ◦ AD = x, BD = y , and observe that ∠ ADC = ∠ CDB = 60 . By Ptolemy’s Theorem, CD = x + y . We have √ 1 3 ◦ 4 = [ ACD ] = AD · CD · sin 60 = x ( x + y ) 2 4 √ 1 3 ◦ 3 = [ BCD ] = BD · CD · sin 60 = y ( x + y ) 2 4 By dividing these equations find x : y = 4 : 3. Let x = 4 t, y = 3 t . Substitute this into the first equation √ 3 2 to get 1 = · 7 t . By the Law of Cosines, 4 2 2 2 2 AB = x + xy + y = 37 t . The area of ABC is then √ 2 AB 3 37 = . 4 7