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

An equilateral triangle

Let $P$ be any point inside of an equilateral triangle $\Delta ABC$. Let $D, E, F$ be the projection from $P$ to $BC, CA, AB$. Let $A', B', C'$ are the projections from $A, B, C$ to the line pass through $P$.

Prove that $PE\times BB'+PF\times CC'=PD\times AA'$

Author: Van Khea, Prey Veng, Cambodia

Geometry, Training

let $D, E, F$ be midpoints of $BC, CA, AB$. Let $(d)$ be any line and let $AA_1, DD_1, EE_1, FF_1$ be distances from $A, D, E, F$ to the line $(d)$.

Prove that  $\overline{AA_1}=\overline{DD_1}+\overline{EE_1}+\overline{FF_1}$

Author: Van Khea, Prey Veng, Cambodia