## 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

## some of problem in equilateral triangle

Given an equilateral triangle $\Delta ABC$. Let $(l)$ is a line tangent to incircle $(I)$. Let $P, Q, R$ be projection points from $A, B, C$ to line $(l)$.

Prove that $IP^4+IQ^4+IR^4=33r^4$

Author: Van Khea, Cambodia

## Excircle

Given an equilateral triangle $\Delta ABC$. Let $P$ be any point lie on incircle.

Prove that $PA^4+PB^4+PC^4=99r^4$

Author: Van Khea, Cambodia

## The equality in equilateral triangle ABC

$\Delta ABC$ is an equilateral triangle. $P$ lie on circumcircle and $Q$ lie on incircle. $m, n, p$ are distances from $P$ and $x, y, z$ are distances from $Q$ to $BC, CA, AB$.

Prove that $\displaystyle \frac{m^2+n^2+p^2}{x^2+y^2+z^2}=2$

Author: Van Khea, Prey Veng, Cambodia