Distributive law for multiplication. ============================================================

Product of a difference and a number. ============================================================

Product of two sums. ============================================================

Product of two differences. ============================================================

a^{2=bx ============================================================}

[a+b] ²=a²+2ab+b² ============================================================

[a-b] ²=a²-2ab+b² ============================================================

[a^{2-b2]=[a-b][a+b] ============================================================}

[a+b+c]2=a²+b²+c²+2ab+2bc+2ca ============================================================

[a+b]²-[a-b]²=4ab ============================================================

Pythogoras' theorem ============================================================

Classical proof of Pythogoras theorem using triangles. ============================================================

a²+b²=c² ============================================================

a²=cp (Similarly you can show that b²=cq ============================================================

a�=p�+h� (Pythagorean theorem), a�=pc=p�+pq (Euklid's theorem), hence h�=pq ============================================================

a�+b�=c� ============================================================

a�+b�=c� ============================================================

c�=(a-b)�+2ab; c�=a�+b� ============================================================

a�+b�=c� ============================================================

a�+b�=c� ============================================================

You use the formula of the area of a trapezium [A=mh, here h=a+b and m=(a+b)/2] (a+b)�/2=c�/2+2*(1/2*ab) or a�+b�=c� ============================================================

c�=a�+b� ============================================================

(a+b)�=c�+4*(1/2ab); a�+b�=c� ============================================================

(a+b)�=a�+3a�b+3ab�+b�

You can see both cubes and the six rectangular parallelepipeds in 3D-view:

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Cube of a difference The formula is (a-b)�=a�-3a�b+3ab�-b�. You convert it to (a-b)�=a�-3ab(a-b)-b� for an illustration. ...... You take the drawing of the formula (a+b)�=a�+3a�b+3ab�+b� from above and replace a by the difference a-b. Then the edges are (a-b)+b with different combinations (on the left). The term (a-b)� is illustrated by the blue cube (on the right).

...... You recieve the blue cube, too, if you take away the three green bodies and the yellow cube from the red cube: (a-b)� = a�-3ab(a-b)-b� = a�-3a�b+3ab�-b� ============================================================