Square is Sum of Two Rectangles

Theorem
If a straight line is cut at random, the rectangle contained by the whole and both of the segments equals the square on the whole.

Proof

 * Euclid-II-2.png

Let $$AB$$ be the given straight line cut at random at the point $$C$$.

Construct the square $$ABED$$ on $$AB$$.

Construct $CF$ parallel to $$AD$$.

Then $$\Box ABED = \Box ACFD + \Box CBEF$$.

Now $$\Box ABED$$ is the square on $$AB$$.

Similarly, from Opposite Sides and Angles of Parallelogram are Equal: Hence the result.
 * $$\Box ACDF$$ is the rectangle contained by $$AB$$ and $$AC$$, as $$AB = AD$$;
 * $$\Box CBFE$$ is the rectangle contained by $$AB$$ and $$BC$$, as $$AB = AD$$.