Linear Algebra Done Right · Problem 6A.34

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · inner-products-and-norms

Problem

Use inner products to prove Apollonius's identity: In a triangle with sides of length

a

b

, and

c

, let

d

be the length of the line segment from the midpoint of the side of length

c

to the opposite vertex. Then

a^2 + b^2 = \tfrac{1}{2}c^2 + 2d^2.