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Let the incircle of triangle $ABC$ be tangent to sides $\overline{BC}$, $\overline{AC}$, and $\overline{AB}$ at $D$, $E$, and $F$, respectively. Prove that triangle $DEF$ is acute. Diagram


I can't prove that the triangles are similar, and we are not given any specific angle of side measurements. I'm stuck, any answer is greatly appreciated.

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  • $\begingroup$ Hint: calculate angle $\widehat{EDF}$ for example. $\endgroup$ – dxiv Sep 22 '16 at 0:25
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$CDE$ is an isosceles triangle, hence $\widehat{EDC}=\frac{\pi-C}{2}$. In a similar way, $\widehat{DBF}=\frac{\pi-B}{2}$, hence: $$ \widehat{FDE}=\pi-\frac{\pi-B}{2}-\frac{\pi-C}{2}=\frac{B+C}{2}$$ and since $B+C<\pi$, $\widehat{FDE}$ is an acute angle. The same applies to $\widehat{FED}$ and $\widehat{DFE}$.

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  • $\begingroup$ What do you mean by $B+C<\pi$? $\endgroup$ – Yuna Kun Sep 22 '16 at 0:57
  • $\begingroup$ @YunaKun: that the amplitude of the $B$ angle, plus the amplitude of the $C$ angle, is less than $180^\circ$ because $ABC$ is a triangle. $\endgroup$ – Jack D'Aurizio Sep 22 '16 at 0:59
  • $\begingroup$ I haven't learned trig, so could you revise your solution so that it doesn't include advanced geometry? Thanks! $\endgroup$ – Yuna Kun Sep 22 '16 at 1:03
  • $\begingroup$ Advanced geometry?! I am just considering the amplitude of some angles. I am unable to produce anything easier than this. $\endgroup$ – Jack D'Aurizio Sep 22 '16 at 1:04
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    $\begingroup$ You may compute the squared lengths of the sides of $DEF$ and apply the converse Pythagorean theorem, if you are happy with that, instead of using that the sum of the angles of a triangle is $180^\circ$. $\endgroup$ – Jack D'Aurizio Sep 22 '16 at 1:08

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