How to prove that $m_a^2+m_b^2=5m_c^2$

Question

I want to prove the formula $$m_a^2+m_b^2=5m_c^2$$ for a right triangle whit c the median of the hypotenuse. I find it this formula in https://en.wikipedia.org/wiki/Median_(geometry) and I want to prove it. If you can, please use easy steps to prove it, I'm 9th class ( I don't know the US educational system). Thank you so much

Answer 1

In any triangle $ABC$ we have $$m_a=\frac{1}{2}\sqrt{2b^2+2c^2-a^2}.$$ Thus, since $m_c=\frac{c}{2}$, we need to prove that $$2b^2+2c^2-a^2+2a^2+2c^2-b^2=5c^2$$ or $$a^2+b^2=c^2,$$ which is obvious.

Answer 2

Using Pythagorean Theorem $$m_a^2=b^2+(\frac{a}{2})^2$$ $$m_b^2=a^2+(\frac{b}{2})^2$$ $$m_a^2+m_b^2=5(\frac{a^2+b^2}{4})$$ $$c^2=a^2+b^2$$ and length of median to hypothenuse is $$2m_c=c$$