## Reflection in triangle 2018

Let $P_0, Q_0, R_0$ be points lies on $B_0C_0, C_0A_0, A_0B_0$.

Let $P_k\in [A_0P_0)$ such that $A_0P_0=P_0P_1=...=P_{k-1}P_k$ and Let $A_k\in [P_0A_0)$ such that $P_0A_0=A_0A_1=...=A_{k-1}A_k$.

The same ways we get $Q_k, B_k$ and $R_k, C_k$.

Prove that $\displaystyle \frac{[A_kB_kC_k]-[P_kQ_kR_k]}{[A_0B_0C_0]-[P_0Q_0R_0]}=2k+1$

Author: Van Khea, Cambodia

## Problem 14/01/18

Let $O$ be circumcenter of an equilateral triangle $\Delta ABC$ and let $O'$ be circumcenter of $\Delta BOC$. Let $(d)$ tangent to $(O')$ at $P$. $M, N$ are projection points from $P$ to $AB, AC$. $B', C'$ are projection points from $B, C$ to $(d)$.

Prove that $PM+CC'=PN+BB'$

Author: Van Khea, Cambodia

## P17, equilateral triangle

Let $D$ be point lie on arc $BC$ of an equilateral triangle $\Delta ABC$  such that $\angle DAC=15^0$. Let $E$ and $F$ be points such $BE//AD$ and $CF//AD$.

Prove that $AD^2+BE^2=2.CF^2$

Author: Van Khea, Cambodia

## Problem 02 (An Equilateral triangle)

$D, E, F$ be midpoints of an equilateral triangle $\Delta ABC$. $T$ be point lie on incircle of $\Delta DEF$.

Prove that $\displaystyle BP+CQ+AR=\frac{3}{2}.AB$

Author: Van Khea, Cambodia

## Problem 08

Line $(d)$ tangent to incircle of an equilateral triangle $\Delta ABC$ at $P$.

Prove that $\displaystyle \frac{AP}{PD}+\frac{AQ}{QE}=1$

Author: Van Khea, Cambodia

## Line tangent to circumcircle

$\Delta ABC$ is an equilateral triangle. Prove that:

$1)$ $PD=PI+PE$

$2)$ $\displaystyle PD^4+PE^4+PI^4=\frac{1}{2}.AB^4$

Author: Van Khea, Cambodia

## right triangle inequality

Let $\displaystyle \frac{AH}{HC}=\frac{7}{3}$.

Prove that $\displaystyle AB^3+BC^3<\frac{3}{4}.AC^3$

Author: Van Khea, Cambodia