In an acute-angled triangle ABC, AB < AC, BD and CE are the altitudes. Prove that $AB^2 + CE^2 < AC^2 + BD^2$
$AB^n + CE^n < AC^n + BD^n$ for positive integers n
I have already found BD < CE and AD < AE but cannot seem to find more information. I already tried using the pythagorean and triangle areas to try and prove it.