Equilateral Triangles, Midpoints, Midlines, Medians, and Geometric Probability

We are coming close to the end of our 14th season of Mathlete Nation’s discovery of mathematics and its mysteries. While playing with a compass used to draw circles, I discovered that with a single line segment, if I set the compass at that segment’s distance, and draw arcs from each endpoint of the segment, the intersection creates a third point. Together they form the vertices of an equilateral triangle (all sides have equal length and all angles have equal measure at 60 degrees). 

If we locate the midpoint of each side of the triangle (the half-way point between vertices), and connect those midpoints with segments we call “midlines,” a new equilateral triangle is formed with an area 1/4 of the original equilateral triangle. If we connect a vertex to its opposite midpoint, we form a new segment we call a median. This median in turn locates the midpoint of each midline and we repeat the process until we can go no further.

The Mathletes discovered that we can draw only 7 inscribed equilateral triangles until we reach the centroid (which is the center of gravity of each centered equilateral triangle). By labeling each point from A to P, we are able to have a common language discussion about each of the dozens of triangles formed. There are equilateral triangles, right triangles, obtuse triangles, kites, and many other forms of polygons formed therein.

The attached pdf has 22 challenges where we look for area relationships or ratios of one triangle of another. For example, by looking for the ratio of triangle DEF/ triangle ABC, using the other pdf for shading (there are 24 pictures of the equilateral triangle diagram), shade in triangle DEF and outline triangle ABC. You will easily see that triangle DEF is 1/4 of the area of triangle ABC. Use a different picture to shade each numerator triangle and outline each denominator triangle. The more difficult ratios will require simple multiplication of triangles within triangles. The most complex ratio is triangle NOP/triangle ABC where you have to take the fourth power of 1/4 or 1/4 x 1/4 x 1/4 x 1/4 = 1/256. You can see that if you multiply 1 x 1 x 1 x 1 = 1 and if you multiply 4 x 4 x 4 x 4 = 256.

The pdf also has equilateral triangle dot paper for the Mathletes to create their own diagrams with any shapes. Finally, there is an answer key so they can check their answers.