Let $A_1, A_2, ..., A_n$ lies on circle $(O)$ such that $A_1A_2=A_2A_3=...=A_nA_1$. Prove that:

$\displaystyle \frac{1}{A_1A_2^2}+\frac{1}{A_1A_3^2}+...+\frac{1}{A_1A_n^2}=\frac{n^2-1}{12R^2}, \forall{n\ge 3}$.

Author: VanKhea

Generalization: RagvaloD

## ទ្រឹស្ដីបទមេនេឡុស និងអនុវត្ត

Let $P, Q, R$ be collinears points on the sides $BC, CA, AB$.

Prove that if  $\displaystyle \frac{AR}{AB}-\frac{BP}{BC}+\frac{CQ}{CA}=1$ then $\displaystyle \frac{AB}{AR}-\frac{BC}{BP}+\frac{CA}{CQ}=-2$

Author: Van Khea

## Polygone identity

Let $A_1, A_2, ..., A_k, k\ge 3$ lies on circle $(O)$ such that $A_1A_2=A_2A_3=...=A_kA_1$ and let $P\in (O)$ such that $arc PA_1=m.arc PA_2$. Prove that:

$\displaystyle \frac{1}{PA_1^2}+\frac{1}{PA_2^2}+...+\frac{1}{PA_k^2}=\frac{k^2}{4R^2sin^2(m\pi)}$

Author: Van Khea

Generalization: Tintarn

## Geometry for Test

Given a triangle $\Delta ABC$ such that $AC=\sqrt{2}R$ and $\angle B=45^0$.

Prove that $\displaystyle \frac{1}{BA}+\frac{1}{BC}=\frac{\sqrt{2+\sqrt{2}}}{R}$

Author: Van Khea

## Centroid G of ABC: Author Van Khea

Let $G$ be the centroid of $\Delta ABC$. Let $AG, BG, CG$ cuts $BC, CA, AB$ at $D, E, F$ and cuts circumcircle of $\Delta ABC$ at $X, Y, Z$. Let $YZ, ZX, XY$ cuts $AG, BG, CG$ at $P, Q, R$.

a) Prove that: $\displaystyle \frac{GD}{GX}+\frac{GE}{GY}+\frac{GF}{GZ}=\frac{3}{2}$

b) Prove that: $\displaystyle \frac{GD}{GP}+\frac{GE}{GQ}+\frac{GF}{GR}=3$

Author: Van Khea

## author by vankhea

Let $P$ be any point inside of $\Delta ABC$. Let $AP, BP, CP$ cuts $BC, CA, AB$ at $D, E, F$ and cuts circumcircle of $\Delta ABC$ at $X, Y, Z$.

Prove that: $\displaystyle (\frac{XP}{PA}+\frac{YP}{PB}+\frac{ZP}{PC})(\frac{AD}{DX}+\frac{BE}{EY}+\frac{CF}{FZ})\geq 27$

Author: Van Khea

## Centroid of triangle ABC

Let $D, E, F$ be midpoints of $BC, CA, AB$. Let $AD, BE, CF$ cuts circumcircle of $ABC$ at $X, Y, Z$.
Prove that $\displaystyle \frac{XD^2}{DA^2}+\frac{YE^2}{EB^2}+\frac{ZF^2}{FC^2}\geq \frac{1}{3}$