## 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

## Right triangle inequality

Let $\Delta ABC$ be right triangle which $\angle A=90^0, BC=a, CA=b, AB=c$.

Prove that $a^3+b^3+c^3\ge (2+\sqrt{2}).abc$.

Author: Van Khea, Cambodia

## A problem of square

Let $ABCD$ be square and let $X, Y, Z, T$ be midpoints. For any point $P$ inside of square $ABCD$, Prove that:

$\sum PA^2-\sum PX^2=a^2$. ( $a=AB$ )

Author: Van Khea, Cambodia

## Square

Let circle $(C)$ inscribe in square $ABCD$. Let $P$ be point lie on circle $(C)$.

Prove that $PA^6+PB^6+PC^6+PD^6=252.r^6$

Author: VanKhea, Cambodia

## The distance from a point to the side of an equilateral triangle

Let $I$ be incenter of an equilateral triangle $\Delta ABC$. Let $P$ be any point and let $D, E, F$ be projection points from $P$ to the sides $BC, CA, AB$.

Prove that $\displaystyle PD^2+PE^2+PF^2=3r^2+\frac{3}{2}.PI^2$

Author: Van Khea, Cambodia

## Equality in Triangle

Let $O$ be circumcenter of an equilateral triangle $\Delta ABC$ and let $P$ be any point in the plan.

Prove that $PA^4+PB^4+PC^4=3R^4+3PO^4+12.PO^2.R^2$

Author: Van Khea, Cambodia