Proof: the distance from a point on an equiaxed hyperbola to the center of the hyperbola is the equal proportion median of its distance from the focus

Proof: the distance from a point on an equiaxed hyperbola to the center of the hyperbola is the equal proportion median of its distance from the focus

Let the hyperbola be x ~ 2 / a ~ 2-y ~ 2 / a ~ 2 = 1, and the square of the distance from any point (x0, Y0) to the center is equal to x0 ~ 2 + Y0 ~ 2, because x ~ 2-y ~ 2 = a ~ 2 and both sides add x ~ 2 + y ~ 2-A ~ 2, then the product of x0 ~ 2 + Y0 ~ 2 = 2x0 ~ 2-A ~ 2 to two focal points is equal to | (ex + a) (ex-a) | because e = root 2, so | (ex0 + a) (ex0 -