Hart’s theorem says

If a triangle be formed by the arcs of three circles, the inscribed and the three escribed circles are all tangent to a new circle or line.

Here “triangle” means a three-sided figure whose sides are portions of a circle. The inscribed circle is the largest circle that can fit inside the three-sided figure.

The “escribed” circles are analogous to the excircles in the previous post: you extend two sides and find a circle that is tangent to the triangle side and the extended side. The difference here being that the side extensions are now circles.

This post will reveal the connection between my two previous posts: one on the Star Trek lemma and one on Pythagorean triples.

In the process of writing the latter, I looked at the Wikipedia article on Pythagorean triples and noticed this curious paragraph.

In every Pythagorean triangle, the radius of the incircle and the radii of the three excircles are positive integers. Specifically, for a primitive triple the radius of the incircle is r = n(m − n), and the radii of the excircles opposite the sides m² − n², 2mn, and the hypotenuse m² + n² are respectively m(m − n), n(m + n), and m(m + n).

The citation for the paragraph above was the book by my former officemate, which led to the post on the Star Trek lemma. The passage in Arthur Baragar’s book that Wikipedia cites is Exercise 15.3.

Let ΔABC be a right angle triangle with sides of integer length. Prove that the inradius r and the exradii r_a, r_b, and r_c are all integers.

I don’t know whether Arthur discovered this theorem, but I’ll call it Baragar’s theorem for this post.

Incircles and excircles

To unpack Baragar’s theorem, let’s start by saying what incircles and excircles are. Incircles are more familiar. The incircle of a triangle is the largest circle that can be inscribed inside the triangle, and the radius of this circle is the inradius.

Since an incircle is an inscribed circle, you might expect an excircle to be a circumscribed circle, but that’s not it. There are three excircles, one for each side. To find the excircle for a side, extend the other two sides and find the circle tangent to the side and the two extensions. The radius of an excircle is its exradius.

Proof

Baragar’s theorem follows directly from Euclid’s formula for Pythagorean triples mentioned in the previous post

and formulas for the inradius r and the exradii r_a, r_b, and r_c.

Here K is the area of the triangle, which in our case is ab/2, and s is the semiperimeter, half the perimeter.

Expressing the radii in terms of m and n gives the values cited by Wikipedia above.

Illustrating the theorem

I’d like to write a Python script to illustrate the theorem, and knowing the radii of the circles help, but we also need to know the centers of the circles.

The center of the incircle is the weighted average of the vertices, with weights given by the lengths of the opposite sides. That is, if the vertices are A, B, and C, and the sides opposite these vertices are a, b, and c, the the incenter is

The centers for the excircles have remarkably similar expressions. For the incenter of the circle opposite a vertex, flip the sign of the corresponding side.

Python code

Putting it all together, here’s an illustrate the theorem.

And here’s the code that produced it. Note that everything in this section works for right triangles in general, not just Pythagorean triangles.

import numpy as np
import matplotlib.pyplot as plt

def connect(A, B):
    plt.plot([A[0], B[0]], [A[1], B[1]], "C0")

def draw_circle(c, r, color):
    t = np.linspace(0, 2*np.pi)
    plt.plot(r*np.cos(t) + c[0], r*np.sin(t) + c[1], color=color)

a, b, c, = 3, 4, 5

A = np.array([0, b])
B = np.array([-a, 0])
C = np.array([0, 0])

s = (a + b + c)/2
K = a*b/2
r = K/s
ra = K/(s - a)
rb = K/(s - b)
rc = K/(s - c)
I = (a*A + b*B + c*C)/(a + b + c)
Ia = (-a*A + b*B + c*C)/(-a + b + c)
Ib = (a*A - b*B + c*C)/(a - b + c)
Ic = (a*A + b*B - c*C)/(a + b - c)

draw_circle(I, r, "C1")
draw_circle(Ia, ra, "C2")
draw_circle(Ib, rb, "C3")
draw_circle(Ic, rc, "C4")

plt.plot([-2*rc, 2*rb], [0, 0], "C0")
plt.plot([0, 0], [-2*ra, 2*rc], "C0")
plt.plot([(-2*ra - b)*a/b, 2*rb], [-2*ra, 2*rb*b/a + b], "C0")

plt.gca().set_aspect("equal")
plt.show()

I was reading an article this evening and saw a footnote to a book by Arthur Baragar [1]. This caught my eye because he was my officemate at UT for a year.

I found his book on Archive.org and was surprised to see “The Star Trek Lemma” in the table of contents. What could this be?

It’s a theorem that goes back to Euclid that applies to an angle formed by connecting a point to two other points on a circle. The theorem says “The measure of an inscribed angle is half the measure of the arc it subtends.” But why call it the Star Trek lemma? Quoting Arthur:

In the spirit of Euclid, we will refer to this theorem as the Star Trek lemma because of the figure associated with the statement of the theorem. … Before Star Trek, as far as I know, this theorem had no name, though some might call it Euclid III.20, which is its proposition number in Euclid’s Elements (Book III, Proposition 20).

Here is my reconstruction of the figure given in the book.

The lemma says that ∠BAC is half of ∠BOC.

[1] Baragar, Arthur (2001), A Survey of Classical and Modern Geometries: With Computer Activities, Prentice Hall