Orthocenters and Altitudes
by
Becky Mohl
1. Constrict any triangle ABC.
2. Construct the Orthocenter H of the triangle ABC.
3. Construct the Orthocenter of triangle HBC.
4. Construct the Orthocenter of triangle HAB.
5. Construct the Orthocenter of triangle HAC.
6. Construct the Circumcircles of triangles ABC, HBC, HAB, HAC.
In the picture above, we can see that circle c2, c3, and c4 all go through the same point H which is the orthocenter of the triangle ABC. The three circles are the circumcirlces of triangles HBC, HAB, and HAC. The last circle c4 is the circumcircle of the original triangle ABC and does not go through point H which is the orhocenter of the triangle ABC. The circumcirlce cannot ever go through the circumcenter because if is the center of the circumcircle.; therefore, this is the reason that the circle c4 does not and will never go through the point H along with the other three circles. However, can the circumcenter ever go through the orthocenter of the triangle since we know that the othocenter can be outside of the triangle. Does this mean that it can lie on the circumcircle? Use the animation to move the points A, B, and C around to see how the figure above changes or does not change.
8. Construct the nine point circles for ABC, HBC, and HAB. Conjecture.
Move the points A, B, and C around and you will see that no matter the shape of the triangle, the nine point circles of the orthic triangles of the given triangle are still congruent.
Animation
Again use the animation to see how the circle changes along with triangle ABC. WIll the nine point circles always be the same? What happens when the triangle is acute, scalene, obtuse, or right?
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Animation