Experts can see for miles; non-experts are walking around in the tall grass and can’t see past their next step. When it comes to geometry, I’m no expert.

John Cook writes:

Let a, b, and c be the sides of a triangle.

Let p be perimeter of the triangle.

Let r be the radius of the largest circle that can be inscribed in the triangle, and let R be the radius of the circle through the vertices of the triangle.

Then all six numbers can be related in one equation:

2prR = abc.

That’s just wack, and I have no intuition on this sort of geometry problem. Math is weird. I can use dimensionality analysis to check that it’s not necessarily wrong, and I could work out a simple example with an equilateral triangle to check that it works for a special case, but I have no intuition that the formula would work in general.

I assume I could prove the above statement using brute-force Cartesian coordinates: label the points as (x,y)_i, i=1,2,3, then the perimeter is sqrt((x1-x2)^2 + (y1-y2)^2) + sqrt((x2-x3)^2 + (y2-y3)^2) + sqrt((x3-x1)^2 + (y3-y1)^2), and there’s gotta be formulas for the centers of the red and blue circles in the above picture, and then it will be easy enough to write the formulas for r and R, and then I’m guessing that if you multiply the factors and rearrange things in some way or another, it will all work out. Given that the statement is true, it has to work out, right?

I’m also guessing there’s some intuition to explain this, some clear derivation.

My point in posting this is not because I care very much about triangles. Rather, I just want to share with you how little understand I have of this area of math. When it comes to probability and statistics, I have a lot of intuition. Not that my intuition is always right, I just have some framework for thinking about probability and statistics problems, some combination of theoretical structures and thousands of examples that I’ve thought about and worked on over the years. With geometry I don’t have that.

This episode gives me some intuition as to how students can often show so little insight into probability and statistics. The students know these topics are important; they just don’t have the experience and modes of thought that can navigate them through the big picture.

Experts (which I am in probability and statistics, but not in geometry, number theory, etc.) can see for miles; non-experts are walking around in the tall grass and can’t see past their next step.