Illustrations of simple formulas

Why  (a+b)² =a²+2ab+b² ?

Let's try to get a geometric interpretation. First, consider two lines of length a and b. Then draw another line of length (a+b). Now in order to compute (a+b)², draw a square having the length of each side equals to (a+b). Then follow the below figure 1 to calculate the total area of the square. 

Fig: 1,(a+b)² =a²+2ab+b²

Why  (a-b)² =a²-2ab+b² ?

First, without loss of generality let's assume that a > b. Then like the previous proof, consider two straight lines with length a and b. Since a > b, draw a straight line with length (a-b). Then consider a square having the length of each side equals to (a-b). Now you need to find out the area of the square.

Fig: 2, (a-b)² =a²-2ab+b²

Using simple illustrations one can show, a²-b²=(a-b)(a+b). Again, without loss of generality, one can assume that a > b. Then draw a square with side a and another square with side b starting from any one of the vertices of the previous square. See the below figure to complete the proof.

Fig:3, a²-b²=(a-b)(a+b)

One can extend the illustration to (a+b+c)² =a²+b²+c²+2ab+2ac+2bc from (a+b)² =a²+2ab+b², just by considering three straight line. 

Fig:4, (a+b+c)² =a²+b²+c²+2ab+2ac+2bc