Straight Line cannot be in Two Planes

Proof

 * Euclid-XI-1.png

Suppose it were possible to have a straight line in more than one plane.

Let a part $AB$ of the straight line $ABC$ be in the plane of reference, and another part $BC$ be in a plane more elevated.

There will then be in the plane of reference some straight line $BD$ continuous with $AB$ in a straight line.

Therefore $AB$ is a common segment of the two straight lines $ABC$ and $ABD$.

Suppose a circle is described with center $B$ and radius $AB$.

Then the diameters would cut off unequal arcs of the circle.

Hence the result.