Assignment 4 "Umbrellas and Other Rainy Day Observations"

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.

Introduction: We are going to explore the nature of a new triangle that can be created from any acute triangle ABC. This construction of this new triangle begins with the construction of the orthocenter of the triangle ABC. We then construct the midpoints of the segments from the orthodenter to each of the vertices. Below is an illustration of this construction:

Click above to see an animation of this picture.

Given:

Any acute triangle ABC

If we can constuct triangle DEF from the midpoints (D, E and F) of the segments connecting the orthocenter (H) to the vertices.

Then, triangle DEF is similar to triangle ABC and congruent to triangle IJK (the medial triangle).

Proof:

Segments HD and HF are half the length of HB and HC, respectively, by definition of midpoint. And, triangle DHF and triangle BHC share angle at H. Therefore, by side-angle-side similarity theorem, the two triangle are similar.

Then segment DF is half the length of segment BC.

Also, since the two triangles are similar and share an angle, their opposite sides are parallel. This follows from equal base angles.

Likewise, segments DE and EF is parallel and helf the length of segments AB and AC, respectively.

Therefore, by side-side-side similarity, triangle DEF is similar to triangle ABC in proportion of 1:2. But, triangle IJK is also similar to triangle ABC and in proportion of 1:2. So, triangles DEF and IJK are congruent!

Click the picture to explore the euler lines for various triangles.

There are many different relationships that can be explored in the above triangles DEF and its orthic, mid-segment triangle. We have already discussed the triangles' similarity. The most obvious observation that can be made would be the placement of the orthocenter in both triangles. Notice that the orthocenter (H) of triangle DEF is coincident to the orthocenter (h) of the orthic, mid-segment triangle. The centroid (g) and the circumcenter (g) of the orthic, mid-segment triangle are colinear to the euler line of DEF. Moreover (c) and (g) are bounded by (C) and (G).

To make some other observations it is advantageous to transform the original triangle, thus allowing for the properties to evolve. The above picture includes an animation that may be helpful.

Did you notice that the euler line for each triangle is in a proportion of 1:2? We can assume that this is a direct consequence of the fact that triangle DEF is a transformation under a size change of magnitude 2.

As the triangle degenerates and the points D, E, and F become colinear, the euler line tends toward an infinite length. This is depicted by enlongating "umbrella" shapes formed where the tip (H) and the handle (C) get further and further apart (hence, our title).