Nine Point Circles

 

Which of the four triangles that we have been talking about do you think this nine point circle was constructed from? In all realty, the picture above is that of all four nine point circles, one for each the four triangles, triangles ABC, HAB, HAC, and HBC. Yes, all four triangles have the same ninepoint circle. The nine points on the circle are numbered so that they can be put into the nine categories listed previously describing what each point represented. Below is that list repeated four times identifying which category each point falls in for the four different triangles being discussed.

For triangle ABC:

For triangle HAB:

For triangle HAC:

For triangle HBC:

All these lists were defined right from the image above and and the definitions of altitudes and orthocenters. A main reason behind why all four triangles have the same nine point circles lies in the fact that two of the three sides of the original triangle ABC become altitudes of triangles HAB, HAC, and HBC.


Return to Altitudes and Orthocenters