The triangle bounded by the polars of the vertices of a triangle 
with respect to a conic is called its polar triangle.
The following table summarizes polar triangles of named triangle conics that correspond
to named triangles.
Another usage of the term applies in the elliptic plane or on a sphere, where the pole of a line is a point that is at an arc length of 
radians from each point of the line,
in the same way that the poles of the Earth are a quarter circle away from the equator.
Two spherical triangles are mutually polar
if each vertex of one is the pole of an edge of the other, and the arc length in
radians of that edge is supplementary to the interior angle at its pole. On a sphere,
the polar triangle lies in the same hemisphere as
the original triangle.
The arc lengths of the principal circumradius of a spherical triangle and the inradius of its polar triangle sum to 
. The principal circumcenter of a spherical triangle is
the incenter of its polar triangle. The altitude from a vertex of a spherical triangle
passes through the pole of the opposite edge. The altitudes of a spherical triangle
and its polar triangle are concurrent at the mutual orthocenter of both triangles.
See also
Chasles's Polar Triangle Theorem, Perspector, Polar Simplex, Self-Polar Triangle, Spherical Triangle
Portions of this entry contributed by Robert A. Russell
Portions of this entry contributed by Floor van Lamoen
Explore with Wolfram|Alpha
References
Coxeter, H. S. M. "The Polar Triangle and the Orthocentre." ยง11.6 in Non-Euclidean Geometry, 6th ed. Washington, DC: Math. Assoc. Amer., p. 223, 1988.
Referenced on Wolfram|Alpha
Cite this as:
Russell, Robert A.; van Lamoen, Floor; and Weisstein, Eric W. "Polar Triangle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PolarTriangle.html