The multiple centers of a triangle

[lpetrich]

Triangles have a Magic Highway - Numberphile - YouTube
A triangle's Euler line.

Medicenter:

Send a line from each vertex to the median / center point on the opposite side.

All three lines will intersect at one point, the medicenter, given by the average of the three vertices' positions. This is also the barycenter of the triangle if it has constant surface density, the center of mass or the centroid.

Orthocenter:

Send a line from each vertex to the opposite side, such that the line is perpendicular to each side.

All three lines will intersect at one point, the orthocenter.

Circumcenter:

Center of the circumscribed circle, the circle that passes through the triangle's vertices.

Incenter:

Center of the inscribed circle, the circle that touches the edges without crossing them.


 Euler line mentions some more triangle-related points that lie it. The medicenter, orthocenter, and circumcenter are on it, but for a scalene triangle (all side lengths unequal), the incenter is not on it.

For an isosceles triangle (two side lengths equal), the Euler line is the line of symmetry, and the incenter is also on it

For an equilateral triangle (all side lengths equal), all these points coincide.

Triangle Centres and the Euler Line (extra footage) - YouTube

Prof. Zvezdelina Stankova, who presented these results, is from Bulgaria.

 

[lpetrich]

I've found this:
 Encyclopedia of Triangle Centers
ENCYCLOPEDIA OF TRIANGLE CENTERS
Lists numerous special points and kinds of center associated with triangles.

 Triangle center lists several of them. The ones in my OP were discovered in antiquity, and several others were discovered in recent centuries. That encyclopedia lists 41,356 of them the last I checked.

What makes a triangle center? The center must be some function of the triangle vertices' locations: C(x1,x2,x3) It must transform like the vertices do, with shifts, rotations, reflections, and scaling, and it must be invariant over interchange of the vertices: 123, 132, 213, 231, 312, 321.

This means that the center can be expressed in barycentric form, as a weighted sum of the vertex coordinates:
C = w1*x1 + w2*x2 + w3*x3

The weights must be functions of the coordinates that are independent of shifts, rotations, reflections, and scaling:
C = w(x1,x2,x3)*x1 + w(x2,x1,x3)*x2 + w(x3,x1,x2)*x3

where coordinate weights w(x1,x3,x2) = w(x1,x2,x3)

To be independent of shifts, w must be a function of x12 = x2 - x1 and x13 = x3 - x1. There are four scalar functions of it that are invariant under rotation:
x12.x12
x12.x13
x13.x13
x12.eps2.x13

where eps2 is the 2D antisymmetric symbol: {{0,1},{-1,0}}

That last one is sign-reversed by reflection, and its square is a combination of the other three:
(x12.eps2.x13)^2 = (x12.x12)*(x13.x13) - (x12.x13)^2
So we omit it.

Of the three remaining ones, two of them are side lengths:
x12.x12 = a3^2
x13.x13 = a2^2

The third one is x1.x1 - x1.x2 - x1.x3 + x2.x3
Expanding all three side lengths:
a1^2 = x2.x2 + x3.x3 - 2*x2.x3
a2^2 = x1.x1 + x3.x3 - 2*x1.x3
a3^2 = x1.x1 + x2.x2 - 2*x1.x2

So
x12.x13 = (1/2)*(a2^2 + a3^2 - a1^2)

That makes the weights functions of the side lengths a1, a2, a3:
w(a1,a2,a3) = w(a1,a3,a2)

The angle sizes are completely determined by the side lengths:
cos(A1) = (a2^2 + a3^2 - a1^2)/(2*a2*a3)
where A1 is the angle at vertex 1.

So side lengths are all the info that we will need for the weights.

That takes care of shifts, rotations, and reflections, leaving scalings.

To take care of that, w must be independent of scaling. Though angle sizes are also independent of scaling, side lengths are not, and this limits what combinations of side lengths are allowed.

BTW, angle sizes determine the side lengths to within scaling, so one could use angle sizes.