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I know that incentre lie on the Euler line for equilateral and isosceles triangle but I found a claim that incentre can lie on the Euler line for obtuse triangle. So, is this claim true?Also does there exist any scalene and acute ( but neither equilateral or isosceles ) triangle for which incentre lies on the Euler line? Finally, if incentre is on Euler line, then is it at a unique location with respect to other centres (orthocentre, circumcentre, centroid)?

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Case 1. Acute triangle.

Let $ABC$ be an acute triangle and $I$, $O$, $H$ are its incenter, circumcenter and orthocenter, respectively. Note that points $I$, $O$ and $H$ are inside triangle $ABC$.

We will prove that $I$, $O$ and $H$ are collinear iff $ABC$ is isosceles or equilateral. Indeed, suppose that $O$, $I$ and $H$ are collinear but $\triangle ABC$ is scalene. Recall that rays $AO$ and $AH$ are symmetric with respect to angle bisector of $\angle BAC$. Hence, angle bisectors of angles $OAH$ and $BAC$ coincide, so $AI$ bisects angle $AOH$. Since $I\in OH$ we have $$ \frac{AO}{AH}=\frac{IO}{IH} $$ due to angle bisector theorem for $\triangle AOH$. Similarly, we obtain that $$ \frac{AO}{AH}=\frac{BO}{BH}=\frac{CO}{CH}=\frac{IO}{IH}. $$ Finally, note that $AO=BO=CO$, so the last equality implies $AH=BH=CH$. Thus, $O$ and $H$ are distinct circumcenters of triangle $ABC$ which is impossible.

Therefore, in acute triangle $O$, $I$ and $H$ are collinear iff $\triangle ABC$ has equal sides.

Case 2. Obtuse (or right) triangle.

Suppose that in $\triangle ABC$ we have $\angle C\geq 90^{\circ}$. In this case we still can apply the previous argument to triangles $AOH$ an $BOH$ (because rays $AO$ and $AH$ are still symmetric with respect to $AI$; the same for rays $BO$, $BH$ and $BI$). Thus, $$ \frac{AO}{AH}=\frac{BO}{BH}=\frac{IO}{IH}. $$ However, it means that $AH=BH$, so $AB=BC$, so triangle $ABC$ is isosceles, as desired.

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  • $\begingroup$ That's perfect but what about obtuse triangles? @richrow $\endgroup$ Aug 1 '20 at 3:10
  • $\begingroup$ Actually, obtuse triangle has two acute angles, so we can apply this argument for $AOH$ and $BOH$ and obtain $AH=BH$ if $\angle C>90^{\circ}$. I will edit my answer later. $\endgroup$
    – richrow
    Aug 1 '20 at 5:08

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