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

Sweet problem

Let $O$ be circumcenter of triangle $\Delta ABC$.

Prove that:  $\displaystyle \frac{DB\times DC}{AB\times AC}+\frac{EC\times EA}{BC\times BA}+\frac{FA\times FB}{CA\times CB}=1$

Author: Van Khea, Cambodia

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