Rt Triangle fun

Hi, just thought I'd share a little math trick I enjoy. You can transform any ol' triangle into a right triangle, keeping the base and perimeter constant. It's possible due to the fact that every triangle can be considered a "snapshot" of an elliptical orbit, and an ellipse always contains a right triangle.

I'll use a "4-5-7" triangle an example.
You can choose any side to be the base - let's go with 4. Start with finding the isoceles triangle, by averaging the two legs: 4-5-7 becomes 4-6-6.

Let's get the 2 roots of the ellipse...
1. the square root of the isoceles leg (rt6), and
2. half the base (2) *divided by* the first root.
So we have 6/rt6 and 2/rt6. Radicals should usually not be in the denominator but we're just gonna square these, to get 36/6 and 4/6.

The difference of these squares is the right triangle's altitude (32/6), the sum of these squares is the hypotense (40/6), and the original base is 4 (or 24/6).

So, the right triangle corresponding to 4-5-7 is:
b = 4
a = 5 1/3
c = 6 2/3
This is a 3-4-5 triangle (base = 3), scaled up by 4/3.


By the way, equilateral triangles ONLY occur in an ellipse with a 4-3-5 right triangle (base = 4).
:)