What if ABC is an obtuse triangle?
For an obtuse triangle we see the orthocenter moves outside the triangle. The obtuse triangle and the orthocenter seem to switch roles, as the ratios formed are now based from the obtuse vertex of the main triangle rather than the orthocenter. Lets test this theory out. Using the same logic as before: = The area of the exterior triangle = the area of the exterior triangle Dividing the top equation by it’s corresponding pair yields: Equation (p) So, we can see that these same relationships do hold true, but the Orthocenter and the triangle vertex with the obtuse angle switch roles in these relationships. By virtue of the above demonstrations and reasoning, the sum of the rations for the corresponding altitude lengths will also be 2.
