Euclid's Elements Book I

1. In the triangle ABC let the angle ABC be equal to the angle ACB. The claim: the side AB is also equal to the side AC. (Dragging A changes both base angles together; they stay equal.)
2. If AB is unequal to AC, one of them is greater; let AB be greater. From the greater AB let DB be cut off equal to the less AC. (D is drawn as this assumption would place it.)

   Proposition I.3

   Proposition I.3: Given two unequal straight lines, to cut off from the greater a straight line equal to the less.
3. Let DC be joined.

   Postulate 1

   Postulate 1: To draw a straight line from any point to any point.
4. Since DB is equal to AC, and BC is common, the two sides DB, BC are equal to the two sides AC, CB, and the angle DBC is equal to the angle ACB; therefore the base DC is equal to the base AB, and the triangle DBC is equal to the triangle ACB.

   Proposition I.4

   Proposition I.4: If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle, and the remaining angles will be equal to the remaining angles respectively.
5. So the less triangle DBC is equal to the greater ACB, which is absurd, since the whole is greater than the part. Therefore AB is not unequal to AC; it is equal to it.

   Common Notion 5

   Common Notion 5: The whole is greater than the part.

Being what it was required to prove.