美夢成真教甄討論區

平面上有4點 ABPQ A B 是定點 AB=根號3, P,Q 動點

AP=PQ=QB=1 三角形APB面積=m, 三角形PBQ 面積是 n,

求 m^2+n^2 max

1 + 3 - 2√3cosA = PB^2 = 1 + 1 - 2cosQ
cosQ = √3cosA - 1

m^2 + n^2 = [(1/2) * 1 * √3 * sinA]^2 + [(1/2) * 1 * 1 * sinQ]^2
= (3/4)(sinA)^2 + (1/4)(sinQ)^2
= (3/4)[1 - (cosA)^2] + (1/4)[1 - (√3cosA - 1)^2]
= (-3/2)(cosA)^2 + (√3/2)cosA + 3/4
= (-3/2)(cosA - √3/6)^2 + 7/8