Stewart's Theorem

Theorem

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

Let $CP$ be any cevian from $C$ to $P$.

Then:

$a^2 \cdot AP+b^2 \cdot PB=CP^2 \cdot c+AP \cdot PB \cdot c$

Proof

There are two cases to consider:

1. When the cevian is an altitude, the result follows directly from the law of cosines on $\triangle APC$ and $\triangle CPB$.
2. When the cevian is not an altitude, we proceed as follows.

We note from Two Angles on a Straight Line make Two Right Angles that $\angle APC$ and $\angle BPC$ are supplementary.

So one of $\angle APC$ and $\angle BPC$ must be acute and the other must be obtuse.

WLOG let $\angle APC$ be acute and $\angle BPC$ be obtuse.

Then we have:

	\(\displaystyle \)	\(\displaystyle \triangle APC:\)	\(\displaystyle \)	\(\displaystyle b^2\)	\(=\)	\(\displaystyle AP^2 + CP^2 - 2 AP \cdot CP \cdot \cos \left({\angle APC}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \triangle CPB:\)	\(\displaystyle \)	\(\displaystyle a^2\)	\(=\)	\(\displaystyle PB^2 + CP^2 + 2 CP \cdot PB \cdot \cos \left({\angle APC}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	since $\cos \left({180^\circ - \alpha}\right) = -\cos \alpha$ from Sine and Cosine of Supplementary Angles

• We multiply the first by $PB$ and the second by $AP$:

$b^2 \cdot PB = AP^2 \cdot PB + CP^2 \cdot PB - 2 PB \cdot AP \cdot CP \cdot \cos \left({\angle APC}\right)$

$a^2 \cdot AP = PB^2 \cdot AP + CP^2 \cdot AP + 2 AP \cdot CP \cdot PB \cdot \cos \left({\angle APC}\right)$

• Now we add the two equations:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle a^2 \cdot AP + b^2 \cdot PB\)	\(=\)	\(\displaystyle AP^2 \cdot PB + PB^2 \cdot AP + CP^2 \cdot PB + CP^2 \cdot AP\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle CP^2 \underbrace{(PB + AP)}_c + AP \cdot PB \underbrace{(AP + PB)}_c\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle CP^2 \cdot c + AP \cdot PB \cdot c\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

$\blacksquare$

Source of Name

This entry was named for Matthew Stewart.

It is also known as Apollonius's Theorem after Apollonius of Perga.