Assignment 8 Problem 1 for Kanita DuCloux

For this assignment, I will:

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.

7. Conjectures? Proofs?

Observe a few of the constructions.

The orthocenter of triangle HBC is vertex A (of the original triangle), the orthocenter of triangle HAB is vertex C (of the original triangle), the orthocenter of triangle HAC is vertex B (of the original triangle).

Observe what happens when triangle ABC is a right triangle. The orthocenter(H) of triangle ABC is at the right angle, B. The circumcircles of triangles ABC and AHC appear to coincide. The center of the circumcircle of triangle AHC, U, is also the center of the circumcircle of triangle ABC.

The measures show that the radii of all of the circumcircles are equal. Also, the area of triangle ABC and the triangle formed by the connecting the centers of the circumcircles (SUV) are equal. The area of circumcircles are equal which follows from the radii all being equal.

Click orthocenter to construct one.

First construct parallel lines. The following angles are conguent:

<TVB and <VYB, <ZCY and <CYS, since <VYB and <CYS are vertical angles, <TVB is congruent to <ZCY.

<SUZ and <ZTA and <VTB and <TBY, since <ZTA and <VTB are vertical angles, <SUZ is congruent to <TBY.

Since, BC and UV are congruent, trianle ABC is congruent to triangle SUV by ASA.