An equilateral triangle ABC

Let $O$ be circumcenter of an equilateral triangle $\Delta ABC$ and let $P$ be point lie on incircle. Prove that:

$a)$ $PD=PE+PF$

$b)$ $PD^2+PE^2+PF^2=3R^2$

$c)$ $PD^4+PE^4+PF^4=\frac{9}{2}.R^4$

Author: Van Khea, Cambodia

A beautiful problem

Let $P$ be point lie on circumcircle of an equilateral triangle $\Delta ABC$. Let $(D_1D_2)//(BC) ;(E_1E_2)//(CA); (F_1F_2)// (AB)$

Prove that $D_1D_2+E_1E_2+F_1F_2=2.AB$

Author: Van Khea, Cambodia

Geometry, Equilateral triangle

Let $P$ be point lie on circumcircle of an equilateral triangle $\Delta ABC$.

Let $D, E, F$ be projection points from $P$ to $BC, CA, AB$.

$a)$ Prove that $\displaystyle AD^2+BE^2+CF^2=\frac{33}{4}.R^2$

$b)$ Prove that $\displaystyle AD^4+BE^4+CF^4=\frac{369}{16}.R^4$

Author: Van Khea, Cambodia