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Let $D,E,F$ be points on the sides BC,CA,AB respectively of a triangle $ABC$ such that $BD=CE=AF$ and $\angle BDF=\angle CED=\angle AFE$.Prove that $\triangle ABC$ is equilateral.

My attempt - Using sine rule in triange $\triangle AFE$ ,$\triangle EDC$ and $\triangle BFD$ respectively, we have $AE \sin\angle AEF=BF \sin\angle BFD=DC\sin\angle FDC $.

But that does not help much.I am totally unaware what to do.please help.

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    $\begingroup$ Did you mean $\angle AFE$ rather than $\angle AFG$? $\endgroup$ – N. F. Taussig Oct 20 '14 at 14:42
  • $\begingroup$ Yes, sorry I meant $\angle AFE $ $\endgroup$ – Snehil Sinha Oct 20 '14 at 14:51
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    $\begingroup$ Without Lost Of Generality. And $D1$ means the angle at $D$ containing number $1$ in picture. $D1=\min\{\}$ means $D1$ is the smallest angle among those. $\endgroup$ – Quang Hoang Oct 30 '14 at 3:13
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    $\begingroup$ @Quang Hoang And how can we say that∠D1=min{∠D1,∠E1,∠F1}.Can u also plz tell how does it follows from this that ∠C is the largest angle of △ABC. $\endgroup$ – Snehil Sinha Oct 30 '14 at 5:38
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    $\begingroup$ Because $D2=\max\{D2,E2F2\}$, and sum of three angles at $D$ is $180^\circ$. Then look at the three corner triangles for the last argument. Now that I think about it. $\angle C =\angle D2$, and the conclusion follows faster. $\endgroup$ – Quang Hoang Oct 30 '14 at 7:11
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So one has the following settings

enter image description here

WLOG, assume that $A$ is the max angle in $\triangle ABC$. It follows from the picture below that $EF$ is the largest side of $\triangle DEF$.

enter image description here

That means $\angle D2$ is the largest angle in $\triangle DEF$. We have $$\angle D1=\min\{\angle D1,\angle E1,\angle F1\}.$$ This follows that $\angle C$ is the largest angle of $\triangle ABC$, or that $\angle C=\angle A$.

P/S: Please excuse my hand-drawing. I was too lazy for computer graphics.

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  • $\begingroup$ I don't understand one thing. why is $\angle AEF = \angle BFD$? $\endgroup$ – user171358 Oct 25 '14 at 11:06
  • $\begingroup$ @DigitalBrain I didn't use that in the proof. The notation $\angle E1$ stands for $\angle AEF$ (to distinguish from $\angle E2=\angle DEF$). $\endgroup$ – Quang Hoang Oct 25 '14 at 14:54
  • $\begingroup$ But you marked some angles with blue sign. $\endgroup$ – user171358 Oct 25 '14 at 14:59
  • $\begingroup$ @DigitalBrain Sure, you can say that it's bad notations. But that the three angles are equal is a by-product of the problem (the three corner triangles are congruent by ASA). $\endgroup$ – Quang Hoang Oct 25 '14 at 15:22
  • $\begingroup$ Why can one say that $\triangle DEF$ is equilateral? $\endgroup$ – rae306 Oct 26 '14 at 9:00

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