Things which are equal to the same thing are equal to each other.

 

Concept: (1) Things that are equal to the same thing are equal to one another, according to Euclid. (2) When equals are added together, the sums are equal. (3) The remainders are equal when equals are subtracted from equals. (4) Things that coincide with one another are equivalent.

The preceding statement is one of Euclid's axioms.

The following example demonstrates it:

If and, then we can simply say that . b and are equal to the same thing (a here), so they are equal.