Random triangles

At a basic level, a random triangle is simply a triangle whose corners are three random points on a piece of paper.

Mathematically speaking, a few decisions have to be made characterize exactly how the random point selection works. Think of it this way : should every place on the piece of paper be equally likely, or should the middle of the page be more likely to be selected than near the borders?

In this module, we assume that the points are coming from a bivariate normal distribution with unit variances and correlation $\rho$.

Play with random triangles!

The following module generates bunches of random triangles using the bivariate normal distribution with correlation coefficient $\rho$. The red triangles are obtuse, and the green triangles are acute (the likelihood of seeing a right triangle is 0, so it doesn’t get a color.) You can change $\rho$ with the slider under the module. What happens as $\rho$ approaches -1 or 1?

In Professor Strang’s lecture he discusses what the triangles look like in “triangle space”. The basic idea is that every triangle has three angles which sum to $180^{\circ}$, call them $\alpha$, $\beta$, and $\gamma$. Every triangle is therefore represented by a single point in the “triangle space”. Further, the triangle space itself can be broken into four regions.

In the diagram below, the regions of the triangle are colored according to the kinds of triangles which are “zoned” to those regions : the red regions represent obtuse triangles, and the green region represents acute triangles. Notice that as $\rho$ approaches -1 or 1, all of the triangles get pulled towards the corners. Why?