Three Points are Coplanar

Theorem

Let $P_1$, $P_2$ and $P_3$ be points in Euclidean $3$-space.

Then there exists a plane $\PP$ such that $P_1$, $P_2$ and $P_3$ all lie in $\PP$.

That is, $P_1$, $P_2$ and $P_3$ are coplanar.

Proof

Let the straight line $P_1 P_2$ be constructed according to Euclid's first postulate.

Let the straight line $P_2 P_3$ be constructed according to Euclid's first postulate.

Thus $P_1 P_2$ and $P_2 P_3$ intersect at $P_2$.

From Two Intersecting Straight Lines are in One Plane, $P_1 P_2$ and $P_2 P_3$ are coplanar.

Hence as $P_1$, $P_2$ and $P_3$ all lie on either $P_1 P_2$ or $P_2 P_3$, it follows that $P_1$, $P_2$ and $P_3$ are likewise coplanar.

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): coplanar
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): coplanar