Exploring Altitude & Orthocenter
By Princess Browne
Construct any triangle ABC.
2. Construct the Orthocenter H of 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, and HAC.
7. Conjectures? Proofs?
From the construction, we know that the radius of the triangles ABC, ABH, ACH and BCH are congruent. The circumcircles of DABC, DABH, DBCH, and DACH are congruent. The areas of the triangles are also congruent. The orthocenter of DHAC is vertex B, the orthocenter of DHBC is vertex A, and the orthocenter of DHAB is vertex C. Which means that the vertex of triangle ABC is the orthocenter of other triangles.