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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.

Snehil Sinha
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    Did you mean $\angle AFE$ rather than $\angle AFG$? – N. F. Taussig Oct 20 '14 at 14:42
  • Yes, sorry I meant $\angle AFE $ – Snehil Sinha Oct 20 '14 at 14:51
  • I just notice that you see my solution as unclear. I do that on purpose as this is a learning site. Anyhow, let me know if you need clarification. – Quang Hoang Oct 27 '14 at 16:02
  • @Quang Hoang I m really sorry for that but could u plz tell me what does WLOG stand for and what does ∠D1=min{∠D1,∠E1,∠F1} mean.once again I m very sorry – Snehil Sinha Oct 30 '14 at 02:22
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    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. – Quang Hoang Oct 30 '14 at 03:13
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    @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. – Snehil Sinha Oct 30 '14 at 05:38
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    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. – Quang Hoang Oct 30 '14 at 07:11
  • @Quang Hoang One last question, how can we say D2=max{D2,E2,F2}. – Snehil Sinha Nov 04 '14 at 13:45
  • Because as the second picture showed, $EF$ is larger than any other side of $\triangle DEF$. – Quang Hoang Nov 04 '14 at 17:12
  • @Quang Hoang And how is EF larger than the others? – Snehil Sinha Nov 07 '14 at 02:01
  • See the second picture. If we stack up the three corner triangles, then the one with the largest angle among $A,B,C$ has the largest opposite side. – Quang Hoang Nov 07 '14 at 03:41

1 Answers1

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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.

Quang Hoang
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