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?
Best Answer
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.
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 a circle with its radius being half the circumradius of the triangle and with center halfway between the orthocentre and the circumcenter, 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.
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.