"Umbrellas and Other Rainy Day Observations"
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:
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!
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).
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