Let $\triangle ABC$ be an acute-angled triangle; $L$, $M$, $N$ be the feet of perpendiculars respectively from $A$, $B$, $C$ to the opposite sides; $D$, $E$, $F$ be the midpoints of the sides $BC$, $CA$, $AB$ respectively; and $I_1, I_2, I_3$ be the ex-centers of $\triangle ABC$. Then, which of the following is true ?

(A) the ortho-center of $\triangle ABC$ is the in-center of $\triangle LMN$.

(B) The circum-center of $\triangle ABC$ is the ortho-center of $\triangle DEF$.

(C) The in-center of $\triangle ABC$ is the ortho-center of $\triangle I_1 I_2 I_3$.

(D) The centroid of $\triangle ABC$ is the centroid of $\triangle DEF$

No idea how to proceed.

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    $\begingroup$ Try drawing an acute scalene triangle and seeing if any of the four are feasible in your example. If one seems to be true, use the picture to try to work out a proof. The picture doesn't amount to a proof, but it can give helpful clues. $\endgroup$ – rschwieb May 14 '14 at 16:42
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    $\begingroup$ And have you learned any useful facts about these points of interest in your course? $\endgroup$ – rschwieb May 14 '14 at 16:47
  • $\begingroup$ You can Google these things, for e.g 'orthic triangle'. $\endgroup$ – Sawarnik May 14 '14 at 17:14
  • $\begingroup$ Used geogebra. It seems all are true! $\endgroup$ – Ruddie May 16 '14 at 14:59

This image has been obtained using geogebra. It seems all are true!

enter image description here


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