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Given $\triangle ABC$. Let $D$ be the point where the altitude form the $A$ vertex intersect $\overline{BC}$ and the point $E$ is the intersect between the bisector of $\angle ABC$ with $\overline{AC}$. Let $P$ be the point of intersect of $\overline{AD}$ with $\overline{BE}$.

Prove that if $AP=2PD$ and $BP=2PE$, then $\triangle ABC$ is equilateral.

Diagram

This is essentially what I've tried. But I don't know how to continue, I can't find any useful congruences.

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By the Angle Bisector Theorem in $\triangle ABD$, $$\frac{|BA|}{|BD|} = \frac{|PA|}{|PD|} = \frac{2}{1}$$ Therefore, $\triangle ABD$ is a $30^\circ$-$60^\circ$-$90^\circ$ triangle; and, then, so is $\triangle BPD$. This implies that your single-tick-mark segments are congruent to your double-tick-mark segments, so that $\triangle APE \cong \triangle BPD$ (SAS). The conclusion follows.

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  • $\begingroup$ How can you know $\triangle ABD$ is a 30∘-60∘-90∘ triangle ? $\endgroup$ – Keith Apr 15 '14 at 19:08
  • $\begingroup$ @user143201: $\triangle ABD$ is a right triangle whose hypotenuse is twice as long as one of its legs. That only happens for $30^\circ$-$60^\circ$-$90^\circ$ triangles. :) $\endgroup$ – Blue Apr 15 '14 at 19:14
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If AP = 2PD and BP = 2PE, then P is the centroid. ….(#)

From (#), AD is then served both as the median and altitude. This further implies AB = AC. …. (%)

From (#), BE is then served both as the median and angle bisector.

By bisector theorem BA : BC = AE : EC = 1 : 1, meaning that BA = BC. ….(@)

(%) + (@) implies △ABC is equilateral.

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  • $\begingroup$ Why must $P$ be the centroid? You do not know about $CP:PF$. $\endgroup$ – Calvin Lin Apr 15 '14 at 5:05
  • $\begingroup$ If I prove P is the in-center first, will that be sufficient to say P is the centroid? $\endgroup$ – Mick Apr 15 '14 at 5:46
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Hint: Show that $AB \parallel DE$.

Hint: Show that $BD=DE= \frac{AB}{2}$.

Hint: Show that $\angle ABD = 60^\circ$.

Define $C^*$ on $BC$ such that $ABC^*$ is an equilateral triangle.

Hint: Show that $P$ is the centroid of $ABC^*$.

Use the fact that $BP = 2 PE$ to conclude that $C=C^*$.

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  • $\begingroup$ How can I show that $AB||DE$ , Thanks!! $\endgroup$ – Keith Apr 15 '14 at 19:07
  • $\begingroup$ @user143201 Follows directly from the ratio of sides being 2:1. $\endgroup$ – Calvin Lin Apr 15 '14 at 19:08

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