Stewart's Theorem
(Redirected from Apollonius's Theorem)
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Theorem
Let $\triangle ABC$ be a triangle with sides $a, b, c$.
Let $CP$ be a cevian from $C$ to $P$.
Then:
- $a^2 \cdot AP + b^2 \cdot PB = c \paren {CP^2 + AP \cdot PB}$
Proof
\(\text {(1)}: \quad\) | \(\ds b^2\) | \(=\) | \(\ds AP^2 + CP^2 - 2 AP \cdot CP \cdot \map \cos {\angle APC}\) | Law of Cosines | ||||||||||
\(\text {(2)}: \quad\) | \(\ds a^2\) | \(=\) | \(\ds PB^2 + CP^2 - 2 CP \cdot PB \cdot \map \cos {\angle BPC}\) | Law of Cosines | ||||||||||
\(\ds \) | \(=\) | \(\ds PB^2 + CP^2 + 2 CP \cdot PB \cdot \map \cos {\angle APC}\) | Cosine of Supplementary Angle | |||||||||||
\(\text {(3)}: \quad\) | \(\ds \leadsto \quad \ \ \) | \(\ds b^2 \cdot PB\) | \(=\) | \(\ds AP^2 \cdot PB + CP^2 \cdot PB - 2 PB \cdot AP \cdot CP \cdot \map \cos {\angle APC}\) | $(1) \ \times PB$ | |||||||||
\(\text {(4)}: \quad\) | \(\ds a^2 \cdot AP\) | \(=\) | \(\ds PB^2 \cdot AP + CP^2 \cdot AP + 2 AP \cdot CP \cdot PB \cdot \map \cos {\angle APC}\) | $(2) \ \times AP$ | ||||||||||
\(\ds \leadsto \quad \ \ \) | \(\ds a^2 \cdot AP + b^2 \cdot PB\) | \(=\) | \(\ds AP^2 \cdot PB + PB^2 \cdot AP + CP^2 \cdot PB + CP^2 \cdot AP\) | $(3) \ + \ (4)$ | ||||||||||
\(\ds \) | \(=\) | \(\ds CP^2 \left({PB + AP}\right) + AP \cdot PB \paren {PB + AP}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds c \paren {CP^2 + AP \cdot PB}\) | as $PB + AP = c$ |
$\blacksquare$
Source of Name
This entry was named for Matthew Stewart.
It is also known as Apollonius's Theorem after Apollonius of Perga.