Question

Let abc be a triangle such that ac^2 + ab^2 = 5bc^2 . Prove that the medians from the vertices b and c are perpendicular

Answer 1

Answer:

Both are perpendicular

Step-by-step explanation:

Let abc be a triangle such that ac^2 + ab^2 = 5bc^2 . Prove that the medians from the vertices b and c are perpendicular

AC² + AB² = 5BC²

Let say x , y , z are the coordinates of A , B & C respectively

=> (z-x)² + (y-x)² = 5(z-y)²

=> z² + x² -2zx + y² + x² - 2xy = 5(z² + y² - 2yz)

=> 2x² -4y² -4z² -2xy + 10yz - 2xz = 0

=>  x² - 2y² - 2z² -xy + 5yx - xz = 0

=> (x + z - 2y)(x + y - 2z) = 0

=>  ((x + z)/2 - y)((x + y)/2 - z) = 0

(x + z)/2 is Median on AC from B  ,  Y is B

(x + y)/2 is Median on AB from C  ,  Z is C

Their products = 0

=> Both are perpendicular