## An Equilateral problems

Let $D, E, F$ be midpoints of an equilateral triangle $\Delta ABC$. Let $K$ be midpoint of $EF$. Prove that:

$a)$ $\displaystyle PD^2+PE^2=\frac{32}{3}.PF^2$

$b)$ $\displaystyle PD^4+PE^4=\frac{530}{9}.PF^4$

Author: Van Khea, Cambodia

## Geometry, triangle

$\Delta ABC$ is an equilateral triangle.

$E, F$ is midpoints of $CA, AB$ and $CP\perp EF$

Prove that $\displaystyle sin^2(EFP)+sin^2(EPF)=\frac{1}{7}$

Proposed by Van Khea, Cambodia

## 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

## 3 centers collinear

Let $D$ be point lie on $BC$ of $\Delta ABC$. Let $I$ and $J$ be incenters of $\Delta ABD$ and $ACD$. Let $P, Q, R$ be circumcenters of $\Delta AIJ, \Delta BIJ, \Delta CIJ$.

Prove that $P, Q, R$ are collinear.

Author: Van Khea, Cambodia

## An equilater triangle

Given an equilateral triangle $\Delta ABC$. Let $D$ be point on $[BC]$. Let $I$ and $J$ be incenter of $\Delta ABD$ and $\Delta ACD$. Let $K$ be circumcenter of $\Delta AIJ$.

Prove that $\Delta IJK$ is an equilateral triangle.

Author: Van Khea, Cambodia