Question

Please help me Guys....Good Evening​

Answer 1

$\sf \Large Solution!!$

$\sf In\:\Delta ABC$

$\sf \to D\:and\:E\:trisect\:BC.$

$\sf \bullet Let\:AB=y$

$\sf \bullet Let\:BD=x$

$\sf \bullet Let\:DE=x$

$\sf \bullet Let\:BC=3x$

$\sf \bullet Let\:BE=2x$

$\sf \bullet Let\:EC=x$

$\sf By\: Pythagoras\: Theorem$

$\sf \to AD^{2}=AB^{2}+BD^{2}$

$\sf \to AE^{2}=AB^{2}+BE^{2}$

$\sf \to AC^{2}=AB^{2}+BC^{2}$

$\sf LHS=8AE^{2}$

$\sf \to LHS=8(AB^{2}+BE^{2})$

$\sf \to LHS=8(y^{2}+(2x)^{2})$

$\sf \to LHS=8y^{2}+32x^{2}$

$\sf RHS=3AC^{2}+5AD^{2}$

$\sf \to RHS=3(AB^{2}+BC^{2})+5(AB^{2}+BD^{2})$

$\sf \to RHS=3(y^{2}+9x^{2}+5(x^{2}+y^{2})$

$\sf \to RHS=8y^{2}+32x^{2}$

$\sf LHS=RHS$

$\sf \therefore 8AE^{2}=3AC^{2}+5AD^{2}$

$\sf Hence,\:proved.$

Thanks for asking!! :)