Stewart’s Theorem

How does this geometry work? Have you ever thought about that? Well, there are pretty much theorems and axioms that helps Euclidean geometry work. Stewart’s Theorem is one of them. Let’s learn what it is!

Lets’s jump straight into the theorem and the proof.

Theorem

Let a, b and c be the length of the sides of the triangle ABC. Let x be the length of the cevian to the side of length a, a = BC . If the cevian of If the cevian divides the side of length a, into two segments of length m and n, with m adjacent to c and n adjacent to b, then Stewart’s Theorem states that:

(mb² + nc²) / (m + n) — mn = x²

Proof

In this proof, we will be using the Cosine Theorem to handle it. In case you are not familiar with it, please check the “The Law of Cosines” article to understand this proof. Having said that, let’s start the proof.

The plan is straightforward: Say the cevian touches the side BC at point Y. We will apply cosines theorem to the triangles ABY and ACY. Let me note that ∠AYC = 180 -∠AYB. Thus, cos∠AYC = -cos∠AYB.

Let ∠AYC = θ, then ∠AYB = 180-θ. We know by definition of cosines that cosθ = -cos(180-θ). Thus we can write c² = x² + m² + 2mxcosθ instead of writing c² = x² + m² -2xmcos(180-θ). Having written the equations, now we want to get mb² + nc². In order to get that we can expand the first equation with m and the second with n. Luckily, “2nmcosθ” disappear when we sum the expanded equations. At the end we get (mb² + nc²) / (m + n) — mn = x².

We have explained Stewart’s theorem but who is Stewart?

Matthew Stewart was born in the manse at Rothesay, on the Isle of Bute, in1717. His father was Rev Dugald Stewart who was the local minister, and his mother was Janet Bannantyne.

In 1734 Matthew Stewart began to work under Simson at the University of Glasgow. In 1741 he began to study with Maclaurin at the University of Edinburgh. During this time he continued to keep in touch with Simpson on Greek geometry.

In May 1743 Stewart spent several weeks in Glasgow working on mathematics with Simson and assisting him in the production of his text Plane loci.

Stewart was licensed to preach at Dunoon in May 1744, then one year later he became a minister at Roseneath, Dumbartonshire.

Stewart married Marjorie Stewart, who was the only daughter of Archibald Stewart, on 20 May 1750. They had several children who died as infants but one of the called Dugald Stewart survived. Dugald Stewart went on to become a mathematician and philosopher and he contributed to building Scottish school philosophy.

Resources,

1 Matthew Stewart. mathshistory. Web. 28.04.2020. (Link)

2 Matthew Stewart. Wikipedia. Web. 27.04.2020. (Link)

3 Stewart’s Theorem. Wikipedia. Web. 25.04.2020 (Link)

4 Stewart’s Theorem. WolframMathWorld. Web. 26.04.2020. (Link)