The feet of the three altitudes of a triangle form the orthic triangle. The centroid of a triangle is both the average of the vertices and the intersection of its medians. For an arbitrary point in a triangle, the lines from that point to the vertices are known as the cevians. If the cevians are reflected by the angle bisectors they intersect at the isogonal conjugate. The symmedian point of a triangle is the isogonal conjugate of the centroid .
For a triangle with a circumcircle, lines tangent at the vertices form the tangential triangle. Lines through the vertices of the orthic and tangential triangles intersect at the point .
For a triangle , the center is the centroid of the orthic triangle (shown in purple) of . The points , and are collinear.
, , be the side lengths,
, , be the circumradius, inradius and semiperimeter of ,
, , be the exact trilinear coordinates of with respect to and .
A triangle center is said to be "even center" if its barycentric coordinates can be expressed as a function of three variables , , that all occur with even exponents. If the center of a triangle has barycentric coordinates as a constant, it is called a "neutral center" (the centroid is the only neutral center). Conversely, a triangle center is said to be "odd center" if it is neither even nor neutral.
Standard barycentric coordinates of a point with respect to a reference triangle have a sum of 1.