设圆上两点为A(a,b),C(c,d),M(x,y):
a² + b² = 9 (1)
c² + d² = 9 (2)
x = (a + c)/2,a + c = 2x (3)
y = (b + d)/2,b + d = 2y (4)
AP² + CP² = AC²
(a - 1)² + (b - 2)² + (c - 1)² + (d - 2)² = (a - c)² + (b - d)²
简化得:a + c + 2(b + d) - 5 = ac + bd
利用(3)(4):x + 2y - 5 = ac + bd (5)
直角三角形斜边上的中线等于斜边的一半:MP = AC/2
4MP² = AC²
4(x - 1)² + 4(y - 2)² = (a - c)² + (b - d)²
= a² - 2ac + c² + b² - 2bd + d² = (a² + b²) + (c² + d²) - 2(ac + bd)
= 9 + 9 - 2(ac + bd)
利用(5)并整理得M的轨迹:
:(x - 1/2)² + (y - 1)² = 13/4