Question

If lengths of two medians of a triangle are equal then the triangle is isosceles.

If true, enter $1$ else enter $0$

Answer 1

Let the $a,b$ and $c$ be three sides of triangle

Then median is

$M_a=\frac{1}{2}\sqrt{2b^2+2c^2-a^2}$

$M_b=\frac{1}{2}\sqrt{2a^2+2c^2-b^2}$

$\therefore \frac{1}{2}\sqrt{2b^2+2c^2-a^2}=\frac{1}{2}\sqrt{2a^2+2c^2-b^2}$

$\Rightarrow 2b^2+2c^2-a^2=2a^2+2c^2-b^2$

$\Rightarrow -a^2-2a^2=-b^2-2b^2$

$\Rightarrow -3a^2=-3b^2\Rightarrow a^2=b^2$

$\Rightarrow a=b$

Then two sides are equal then triangle is isosceles