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We know that the orthocenter of a triangle is the place where the triangle's three altitudes intersect. But why is it called that?

The *in*center - the intersection of the triangle's three angle bisectors - is the center of the *in*scribed circle.

The *circum*center - the intersection of the triangle's three perpendicular bisectors - is the center of the *circum*scribed circle.

But what is an "orthocircle" (if such exists)?

This document says that two circles could be orthogonal. Well gee, whiz!

This one says an orthocircle has the given center and is orthogonal to the given circle. That doesn't help, because I can make an orthogonal circle to the triangle from many points (maybe even all of them).

And according to Wolframalpha, the name "orthocenter" was invented by people named Besant and Ferrers. But that page doesn't indicate why how came up with it.

And that is literally everything I could find on the word "orthocircle" or on "why is it called orthocenter?".

So why is it called "orthocenter"? And what is the "orthocircle"?

P.S. And how do I italicize part of a word?

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    $\begingroup$ This is just a guess, but I believe it is because orthocenter is formed form dropping three altitudes, which by definition makes an angle of $90^0$ with its opposite base. $\endgroup$ – IAmNoOne Jan 16 '14 at 5:11
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    $\begingroup$ Ortho- because orthogonal means perpendicular, -center because it is a triangle center (which is not necessarily the center of any useful circle). $\endgroup$ – user856 Jan 16 '14 at 5:15
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Ortho means "straight, right". Orthocenter, because it is the intersection of the lines passing through the vertices and forming right-angles with the opposite sides.

There are many circumferences associated to the orthocenter. I don't think (or better said I haven't heard of) any of them called orthocircle.

Nevertheless, the orthocenter is the center of the circle that passes through the vertices of the anticomplementary triangle (it is the circumcenter of the anticomplementary triangle). The anticomplementary triangle has sides passing through the vertices of the triangle, parallel to the opposite sides.

Anticomplementary triangle

Equivalently the original triangle has vertices at the midpoints of the sides of the anticomplementary triangle. Maybe this could be called the orthocircle.

Also, if you take the circumcircle of the triangle and contract it with ratio $1/2$ with center the orthocenter, we get the nine-point circle. This circle passes through the feet of the altitudes, the mid-points of the sides, and the mid-points between the orthocenter and the vertices.

Nine-point circle

In the picture above, $H$ is the orthocenter, $O$ is the circumcenter, and $G$ is the center of the nine-point circle, which is halfway between $H$ and $O$.

The nine-point circle is $1/4$ times contraction of the circumcenter of the anticomplementary triangle, again a contraction with center at the orthocenter. Equivalently, the circumcenter of the anticomplementary triangle is the $2$ times expansion of the circumcenter of the original triangle, by a contraction again with center at the orthocenter.

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  • $\begingroup$ In the nine-point circle, where do you get the points labelled Ea, Eb and Ec ? $\endgroup$ – Kantura Sep 16 '17 at 21:01
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    $\begingroup$ @Kantura I guess, by looking at the figure, Ea is midpoint of AH, Eb is midpoint of BH and Ec is midpoint of CH. $\endgroup$ – Mallikarjun M Nov 12 '19 at 10:52

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