Investigation of Orthocenter and Altitudes


Nathan Wisdom


The ORTHOCENTER (H) of a triangle is the common intersection of the three lines containing the altitudes. An altitude is a perpendicular segment from a vertex to the line containing the opposite side of the triangle.

We begin this investigation with the following set of constructions

1. 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.




GSP File

Conjectures and Observations



NOW SOLVE This... Prove that triangle ABC is congruent to triangle EFG


The internal angle bisectors of triangle ABC are extended to meet the circumcircle at points L, M, and N, respectively. Find the angles of triangle LMN in terms of the angles A, B, and C.


GSP File



Does your result hold only for acute triangles? From The GSP construction it is easy to see that the result holds not only for acute triangles.

Given triangle ABC. Construct the Orthocenter H. Let points D, E, and F be the feet of the perpendiculars from A, B, and C respectfully. Prove:

1=(HF/CF)+ (HD/AD)+ (HE/BE)

Now Solve This... Prove that

2=(AH/AD)+ (BH/BE)+ (CH/CF)