Square of Sum

Theorem

 * $\forall x, y \in \R: \left({x + y}\right)^2 = x^2 + 2 x y + y^2$

Algebraic Proof
Follows from the distribution of multiplication over addition:


 * $(x + y)^2 = (x + y) \cdot (x + y) = x \cdot (x + y) + y \cdot (x + y) = x \cdot x + x \cdot y + y \cdot x + y \cdot y = x^2 + 2xy + y^2$

More succinctly, it follows directly from the Binomial Theorem:
 * $\forall n \in \Z_+: \left({x+y}\right)^n = \sum_{k=0}^n {n\choose k}x^{n-k}y^k$

putting $n = 2$.

Geometric Proof
As Euclid put it:


 * "If a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments."


 * Euclid-II-4.png

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

Construct the square $ADEB$ on $AB$ and join $DB$.

Construct $CF$ parallel to $AD$ through $C$.

Construct $HK$ parallel to $AB$ through $G$.

From Parallel Implies Equal Alternate Interior Angles, $\angle CGB = \angle ADB$.

We have that $BA = AD$.

So from Isosceles Triangles have Two Equal Angles, $\angle ADB = \angle ABD$.

So $\angle CGB = \angle CBG$, and so from Triangle with Two Equal Angles is Isosceles, $BC = CG$.

From Opposite Sides and Angles of Parallelogram are Equal, we have $CB = GK$ and $CG = KB$ and so $CGKB$ is equilateral.

Now, since $CG \| BK$, from Parallel Implies Supplementary Interior Angles we have that $\angle KBC$ and $\angle GCB$ are supplementary.

But as $\angle KBC$ is a right angle, $\angle BCG$ is also a right angle.

So from Opposite Sides and Angles of Parallelogram are Equal, $\angle CGK$ and $\angle GKB$ are also right angles.

So $CGKB$, described on $CB$. is right-angled, and as it is equilateral, by definition it is square.

For the same reason, $HGDF$ is also square, and described on $HG$, which equals $AB$.

So $HGDF$ and $CGKB$ are the squares on $AC$ and $CB$.

Now we have from Complements of Parallelograms are Equal that $\Box ACGH = \Box GKEF$.

But $\Box ACGH$ is the rectangle contained by $AC$ and $CB$, as $CB = GC$.

So $\Box GKEF$ is also equal to the rectangle contained by $AC$ and $CB$.

But the squares $HGFD$ and $CBKG$ are equal to the squares on $AC$ and $CB$.

So the four areas $HGFD$, $CBKG$, $ACGH$ and $GKEF$ are equal to the squares on $AC$ and $CB$, and twice the rectangle contained by $AC$ and $CB$.

But $HGFD$, $CBKG$, $ACGH$ and $GKEF$ are also equal to the square on $AB$.

Hence the result.