Stewart's Theorem
Idea · Intermediate
Geometry
"A man and his dad put a bomb in the sink" is a well-known mnemonic for Stewart's Theorem.
Let
Since B, D, and C are collinear,
Apply the Law of Cosines to triangle ABD:
Apply the Law of Cosines to triangle ACD:
Multiply the first equation by n and the second by m:
Adding the equations eliminates the cosine terms:
Since
we obtain
This is Stewart's Theorem.
When AD is a median,
Let the median length be m_a. Stewart's Theorem gives
Therefore,
This is Apollonius's Theorem.
Since the triangle is right-angled at A,
The median to the hypotenuse of a right triangle is half the length of the hypotenuse.
By the Angle Bisector Theorem,
Applying Stewart's Theorem,
Dividing by a and simplifying,
Since
we obtain
In triangle ABC, D lies on BC, and
Find
Let
By Stewart's Theorem,
Simplifying,
Therefore,
and hence