How many "centres" are there of a triangle

One of my students wrote the centre of a triangle is (r1 + r2 + r3)/3. Well that is one definition, the centroid or centre of mass if it is has uniform density.

I immediately thought also of the

• Circumcentre, the centre of the circumscribing circle


• Incenter, the centre of the inscribing circle


• Centroid, where the lines joining the mid point of each side to the opposite vertex meet.

Then some googling found me Jim Loy's page listing a few others

• Orthocenter where the three altitudes of a triangle meet.


• Gergonne point where the lines connecting the tangent points of the incircle to the opposite vertices meet


• Fermat point (also called the isogonic center or the Rorricelli point) which minimizes the sum of the distances from the three vertices.

and three more as well! Then I found an even better page Geometry step by step from the land of the Incas(!) Listing 16 different kinds of centres of a triangle (with pretty colour pictures).