Thursday, October 12, 2006

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).


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