Geometry • Topic 16

Stewart's Theorem

Stewart's Theorem relates the length of a cevian (a line from a vertex to the opposite side) to the side lengths. It is essential for finding lengths of medians and angle bisectors.

Statement

Theorem (Stewart's Theorem)

Let

D

be a point on side

BC

△ A BC

. Let

A D = d

B D = m

C D = n

, and side lengths be

a, b, c

(where

a = m + n

). Then:

b^{2} m + c^{2} n = a (d^{2} + mn)

Mnemonic: "man + dad = bmb + cnc"

$man$ : $m \cdot a \cdot n$ (Wait, usually written $d^{2} a + mna = b^{2} m + c^{2} n$ ).
Common Mnemonic: $d a d + man = bmb + c n c$ ( $d \cdot a \cdot d + m \cdot a \cdot n = b \cdot m \cdot b + c \cdot n \cdot c$ ).

Applications

1. Length of a Median

A D

is a median (

m = n = a /2

), Stewart's simplifies to Apollonius' Theorem:

2 (m_{a}^{2} + (a /2)^{2}) = b^{2} + c^{2}

m_{a} = \frac{1}{2} 2 b^{2} + 2 c^{2} - a^{2}

2. Length of an Angle Bisector

A D

is an angle bisector, we can combine Stewart's with the Angle Bisector Theorem (

m / n = c / b

) to get:

d^{2} = b c - mn

A D^{2} = b c (1 - (\frac{a}{b + c})^{2})

Practice Problems

Exercise (Problem 1)

△ A BC

A B = 5

A C = 8

, and

BC = 10

. Find the length of the median from

A

Exercise (Problem 2)

Calculate the length of the angle bisector

A D

△ A BC

if sides are 4, 5, 6.

Exercise (Problem 3)

Prove that the sum of the squares of the lengths of the medians is

3/4

the sum of the squares of the lengths of the sides.