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It seems well known that the incenter of a triangle lies on the the Euler line if and only if the triangle is isosceles (or equilateral, but that is trivial). Searching the internet, I could not find a simple geometric proof of this fact. Can anyone provide such a proof? Also, when the incenter lies on the Euler line, does it do so in a set location? (For example, we know the centroid is a third of the way from the circumcenter to the orthocenter on the Euler line, does the incenter satisfy any nice ratios like that?)

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Have you tried various extreme forms of triangle to see what the possibilities are. It depends which relationships you are studying, but what would you have to do, do you think, to get an extreme position of the incentre in relation to the other centres? I think there are plenty of resources out there which would enable you to conduct some investigations (one extreme is the equilateral case, where almost any obvious centre is the same). Note that even in this "trivial" case there are centres outside the triangle which give circles tangent to the three sides. – Mark Bennet Jan 8 '12 at 21:58
up vote 4 down vote accepted

Here's a nice proof by contradiction.

Let the incenter $I$ lie on the Euler line of $ABC$.

It is known that orthocenter $H$ and circumcenter $O$ are isogonal conjugates, i.e. $AI$ is the bisector of angle $HAO$.

So, (if the point $A$ does not lie on Euler line) $HA/AO=HI/IO$ (angle bisector theorem). Also $HB/BO=HC/CO=HI/IO=HA/AO$. And we know that all points $X$, such that $YX/ZX=const$, lie on a circle with center on the line $YZ$ (Appolonius circle)

So, $A, B, C$ and $I$ lies on the same circle, and that cannot be true. We have assumed that all of points $A, B, C$ don't lie on the Euler line, so, one of them lies on Euler line and that means $ABC$ is isosceles.

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Wow, it's been a while since I've thought about this problem. Thank you for supplying this wonderful proof, it is exactly what I was looking for. – EuYu Sep 6 '13 at 17:15

One approach could be to use trilinear coordinates and show that the incentre at $1:1:1$ is usually not collinear with for example the circumcentre at $\cos A :\cos B :\cos C$ and the orthocentre at $\sec A :\sec B :\sec C$ by looking at the determinate

$$\begin{vmatrix}1&1&1\\ \cos A &\cos B &\cos C\\ \sec A &\sec B &\sec C\end{vmatrix}$$

which is non-zero unless at least two of $A$, $B$ and $C$ are equal.

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Thank you for the answer. I am primarily interested in a geometrical proof without appealing to coordinates or analytic geometry. – EuYu Jan 9 '12 at 2:40

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