Assignment 4: Similar Triangles
I set out to do problem #8, but I would like to report on a variation.
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.
Instead of letting H be the Orthocenter of ABC, I started with the circumcenter. Since I use C in my triangle ABC, I refer to the circumcenter by H.
I then constructed the circumcenter H and the segments HA, HB, and HC. I constructed the midpoints of HA, HB, and HC, and I connected the midpoints to form a triangle.
It appears that triangle ABC is similar to triangle DEF. Is it?
Claim: Triangle ABC is similar to triangle DEF
Proof (Using Classical Euclidean Geometry):
1. Angle AHC = Angle DHF 1. They are the same angle.
2. (AH)/(DH) =(HC)/(HF)=2 2. D and F are midpoints of AH and HC, respectively.
3. Triangle AHC is similar to Triangle DHF 3. SAS
4. Angle HAC = Angle HDF and 4. Corresponding angles of similar triangles.
Angle HFD = Angle HCA
A similar argument shows that triangle ABH is similar to triangle DEH and triangle BHC is similar to triangle EHF.
Thus, by corresponding parts of similar triangles, angle BAH = angle EDH, angle ABH = angle DEH, angle HBC = angle HEF, and angle BCH = angle EFH.
Thus we have,
Therefore, triangle ABC is similar to triangle DEF by AAA.
Note that this proof will hold even when H is not inside of triangle ABC.
In fact, the proof will hold even when H is not the circumcenter. That gives the following generalization:
Claim: Given a triangle ABC and any point H, construct the segments HA, HB, and HC. Then construct the midpoints of HA, HB, and HC and construct a triangle with these three midpoints as vertices. Then the resulting triangle will be similar to triangle ABC with one forth of the area.
Explore this yourself with Geometer's Sketchpad.
I will use bold letters and endpoints to indicate vectors. For example BA is the vector from B to A.
Proof (Using Vector Algebra):
By vector addition, we have
BA + AH = BH and ED + DH = EH
BA = BH - AH and ED = EH - DH
Since E and D are the midpoints of BH and AH respectively, we have