Assignment 4Centers of a Triangle

**8. 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.**

Prove that triangle EFG is similar to triangle ABC.EG = 1/2 AC, EF = 1/2 AB, FG = 1/2 BC since E, F, and G are midpoints of HA, HB, and HC respectively.

EG/AC = EF/AB = FG/BC = 1/2.

Therefore, triangle EFG is similar to triangle ABC since the ratios of the corresponing sides are equal (1/2).

Prove that triangle EFG is congruent to triangle TUS, which is the medial triangle of the same acute triangle ABC.EG = 1/2 AC, and so is ST. -> EG = ST

EF = 1/2 AB, and so is UT. -> EF = UT

FG = 1/2 BC, and so is SU. ->FG = SU

Therefore, triangle EFG is congruent to triangle TUS as required.