Chasles' Relation

Theorem

Let $\EE$ be an affine space.

Let $p, q, r \in \EE$ be points.

Then:

$\vec {p q} = \vec {p r} + \vec {r q}$

Proof

We have:

\(\ds \vec {p r} + \vec {r q}\)

\(=\)

\(\ds \paren {r - p} + \paren {q - r}\)

Definition of Vector in Affine Space

\(\ds \)

\(=\)

\(\ds \paren {r + \paren {q - r} } - p\)

Definition of Affine Space: axiom $(\text A 3)$

\(\ds \)

\(=\)

\(\ds q - p\)

Definition of Affine Space: axiom $(\text A 1)$

\(\ds \)

\(=\)

\(\ds \vec{p q}\)

Definition of Vector in Affine Space

$\blacksquare$

Source of Name

This entry was named for Michel Chasles.