In this post we summarise some of the points of a triangle that maximise or minimise various functions on the plane. This is on a similar theme to an earlier post on optimisation problems given a point inside a given angle. Many of the results are proved in .
Firstly we introduce some notation and terminology. Let be a given triangle with side lengths and let be a variable point in the plane. Let be the distances from to the sides respectively. The feet of the perpendiculars from to the sides of form a pedal triangle. If meet sides in points , then is known as the Cevian triangle of .
Functions involving distances to the vertices
- : is the Fermat point of . If the largest angle is 120 degrees or more, the Fermat point is at the vertex of this angle, otherwise we can construct it by drawing an equilateral triangle external to , then finding the intersection of and the circumcircle of .
In this figure one can show that .
- : if is not obtuse, is the circumcentre of . Otherwise it is the intersection of the longest side of with the perpendicular bisector of its middle-length side.
- : is the orthocentre of , with minimum value given by four times the area of .
- : is the centroid of with minimum value given by .
- : is the incentre of with minimum value given by .
- : This generalises the previous two expressions: is the point described by the position vector with minimum value equal to .
- : is the orthocentre of with minimum value given by .
Functions involving distances to the sides
- : is the vertex of the largest angle of .
- : is the centroid of .
- : is the Mittenpunkt of , given by the position vector . This and the previous fact can be proved using Lagrange multipliers using the constraint that is constant (equal to twice the area of ). The Mittenpunkt is the point of intersection of lines joining the triangle’s excentres (formed by exterior angle bisectors) and midpoints.
- : is the incentre of (IMO shortlist 2001)
Functions involving the pedal triangle
- maximum area of pedal triangle: is the circumcentre of .
- minimum perimeter of pedal triangle: is the orthocentre of .
- minimum pedal radius (radius of circumcircle of pedal triangle): is the incentre of . See  for more details.
Functions involving the Cevian triangle
- maximum area of Cevian triangle: is the centroid of with maximum area given by a fourth of the area of .
 T. Andreescu, O. Mushkarov, L. Stoyanov, Geometric Problems on Maxima and Minima, Birkhäuser, 2006.
 V. Naik, Optimization methods in planar geometry, available at http://www.cmi.ac.in/~vipul/olymp_resources/olympiadarticles/geometricoptimization.pdf