This is the writeup of Assignment #4 
Brian R. Lawler

EMAT 6680 
12/12/00

The Problem
Take an acute triangle ABC. Construct H and the segments HA, HB, and HC. Construct the midpoints of HA, HB, and HC. Connect the midpoints to form a triangle. Prove that this triangle is similar to triangle ABC and congruent to the medial triangle. Construct G, H, C, and I for this triangle. Compare.
(Note: throughout writeup below, click on images to download original file to explore on your own.)
Triangle ABC with segments HA, HB, and HC and the triangle formed by it's midpoints, triangle FED, is as appears below.
To prove triangle ABC is similar to triangle FED, I will show that two pairs of angles are congruent. Two triangles with two pairs of congruent angles are similar.
Begin by constructing line "l" parallel to line BA and through point H. Since line HA is a transversal, angle HAB is congruent to angle HFE. With a similar construction of a line "m", it is evident angle HAC is congruent to angle HFD. Thus, since angle CAB is congruent to angle HAC + angle HAB and angle DFE is congruent to angle HFD + angle HFE, it must be that angle CAB is congruent to angle DFE.
A similar argument will show that angle CBA is congruent to angle DEF. When two triangles have two pairs of congruent angles, they are similar. Therefore triangle ABC is similar to triangle FED.
Next I sketched the medial triangle of ABC. It is labeled triangle LMN below.
To prove triangle FED is congruent to triangle LMN, I will show first that all angles are congruent. Then I will establish the corresponding sides are equal in length. Two triangles with congruent angles and one side length congruent are congruent.
The picture below shows the Euler line colorcoded to match each triangle. Note that the Euler Line for triangle ABC is hidden under the other two. Additionally, I've changed the fonts to keep all triangles labeled equivalently. The original triangle ABC and it's Euler Line are labeled with underlines. The medial triangle LMN and it's Euler Line are labeled with outlined letters. And the triangle formed at midpoints to H and it's Euler Line are labeled with italics.
Some observations:
PROOF OF THE ABOVE IS LEFT TO YOUR FANCY!
Comments? Questions? email me at blawler@coe.uga.edu 
Last revised: December 28, 2000 