The Mathematics of the Triforce
An Exploration of the Medial Triangle
This write-up is based on a prompt from the Exploration 4 webpage:
Take any triangle. Construct a triangle connecting the three midpoints of the sides. This is called the MEDIAL triangle. It is similar to the original triangle and one-fourth of its area. Construct G, H, C, and I for this new triangle. Compare to G, H, C, and I in the original triangle.
For those of you who are as into video gaming as I have, you have probably been faced with the medial triangle plenty of times without even realizing it. For those of you who play Zelda, this is the shape of the Triforce symbol. For those of you who are not familiar with the Zelda games, here is a quick lesson:
This symbols actually represents the three crests from the game: power, wisdom, and courage. Whoever touches this sacred relic will be granted a wish. But that's not the point of this write-up. It's just an interesting "real-life" application of this concept.
I will begin with constructions of the medial triangle for various types of triangles:
It is easy to see that the medial triangle is 1/4 the area of the original triangle when dealing with an equilateral triangle (the first triangle), but it is a little harder to see this characteristic in other types of triangles. Therefore, how can one tell if the medial triangle actually is 1/4 the area? We could, of course, simply take the areas of the two triangles and see that this is true. However, why is this so?
A formal proof is not necessary. The medial triangle is formed by the midpoints of the sides of a triangle. This means that we are reducing the base by half and the height by half. This reduces the entire area by a factor of 4. In this way, the medial triangle is always similar to the original triangle. One can see this visually by taking the medial triangle and rotating it 180 degrees and displaying it beside the original triangle:
Therefore, the medial triangle is similar to the original and has a ratio of area with the original triangle of 1:4.In fact, the medial triangle splits the original triangle into four equal triangles that are all similar to the original triangle, which makes sense because the medial triangle is 1/4 of the area. Therefore, the other three triangles formed should also be 1/4 the area so that their total areas will be the whole area of the original triangle.
Knowing these characteristics of the medial triangle, my intuition is that the centers of the medial triangle will simply be the reverse of the original triangle and scaled down by 4 to fit into the medial triangle's area (except for the centroid, which should be the same since it is formed from the medians and would thus result in the same point). This is because the orientation of the medial triangle is a rotation of 180 degrees from the orientation of the original triangle. To see if my intuition is correct, I will first construct the centers for my given original triangle:
I will now construct each center one by one and examine their relationship with the original centers, beginning with the centroid:
As stated before, the centroid of the medial triangle is the same as the original triangle. This is because this point is formed from the intersections of the medians. Since the medial triangle is formed by the midpoints of the sides of the triangle, it makes sense that this point is the same because the medians for the medial triangle lie on the medians for the original triangle:
Next, I will construct the orthocenter of the medial triangle:
As you can see on the above image, there are still only four points because the new orthocenter is the same as the first triangle's circumcenter. Why is this? Well, we can infer that the circumcenter of a triangle is equidistant from all three of the vertices. Furthermore, the orthocenter of a triangle is the intersection of the altitudes. What are altitudes? They are created by extending a segment from the vertex of a triangle to the opposite side and creating a 90 degree angle. By creating the medial triangle, we are rotating the triangle 180 degrees and shrinking it down by a factor of 4. In this way, we are taking the triangle causing the orientation of the sides and angles to be the opposite of what they were in the original triangle. Therefore, the intersection of the altitudes of the medial triangle becomes the point where the vertices of the original triangle are equidistant. In essence, the orthocenter of the medial triangle is dependent on the aspects for the circumcenter for the original triangle. That is why they are the same.
What about the incenter?
This point does not coincide with the original incenter, nor does it share any of the other points. However, there is a special characteristic for this point. Let's think about how the incenter is found. It is the center of a circle constructed inside of the triangle. Let's construct these circles for both the original triangle and the medial triangle:
How are these two circles different? We have already stated how the medial triangle is 1/4 the area of the original triangle, and the two triangles are similar. Therefore, we can safely say that the area of the incenter for the medial triangle is 1/4 that of the incircle for the original triangle. But what about the positions of the circles and their centers? What would happen if we constructed the diameters of both the original triangle and the medial triangle?
We find that the diameter of the medial triangle incircle is exactly 1/4 that of the original triangle's incircle, which supports our statement about the areas of the circles being in a 1:4 ratio. But how does this help us find the location of the medial triangle's incenter? Well, we also know that the two triangles are similar. We could construct a line segment from the one of the vertices of the original triangle to its corresponding vertex of the medial triangle. We can then find the distance from the original vertex to the incenter. We can then scale down this measurement by 1/4 and travel along the same segment from the corresponding vertex of the medial triangle to find the location of the new incenter.
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