Euclid's Elements Book I

1. Let ABC be a triangle. The claim: in the triangle ABC any two sides taken together are greater than the remaining one; for example BA, AC are greater than BC. (Try dragging the vertices.)
2. Let BA be produced to the point D, and let AD be made equal to CA.

   Postulate 2Proposition I.3

   Postulate 2: To produce a finite straight line continuously in a straight line.

   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 DA is equal to AC, the angle ADC is equal to the angle ACD.

   Proposition I.5

   Proposition I.5: In isosceles triangles the angles at the base are equal to one another, and, if the equal straight lines be produced further, the angles under the base will be equal to one another.
5. But the angle BCD is greater than the angle ACD; therefore the angle BCD is greater than the angle ADC.

   Common Notion 5

   Common Notion 5: The whole is greater than the part.
6. In the triangle DCB the angle BCD is greater than the angle BDC, and the greater angle is subtended by the greater side; therefore DB is greater than BC.

   Proposition I.19

   Proposition I.19: In any triangle the greater angle is subtended by the greater side.
7. But DB is equal to BA, AC together; therefore BA, AC are greater than BC. Similarly AB, BC are greater than CA, and BC, CA are greater than AB.

Being what it was required to prove.