Euclid's Elements Book I

1. In the triangle ABC let the square on one side BC be equal to the squares on the sides BA, AC. The claim: the angle BAC is right. (Try dragging A, B or C.)
2. From the point A let AD be drawn at right angles to the straight line AC, let AD be made equal to AB, and let DC be joined.

   Proposition I.11Proposition I.3Postulate 1

   Proposition I.11: To draw a straight line at right angles to a given straight line from a given point on it.

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

   Postulate 1: To draw a straight line from any point to any point.
3. Since DA is equal to AB, the square on DA is equal to the square on AB. Let the square on AC be added to each; then the squares on DA, AC are equal to the squares on BA, AC.

   Common Notion 2

   Common Notion 2: If equals be added to equals, the wholes are equal.
4. Since the angle DAC is right, the square on DC is equal to the squares on DA, AC; and the square on BC is equal by hypothesis to the squares on BA, AC. Therefore the square on DC is equal to the square on BC, so that DC is equal to BC.

   Proposition I.47Common Notion 1

   Proposition I.47: In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.

   Common Notion 1: Things which are equal to the same thing are also equal to one another.
5. Since DA is equal to AB and AC is common, and the base DC is equal to the base BC, the angle DAC is equal to the angle BAC. But the angle DAC is right; therefore the angle BAC is also right.

   Proposition I.8

   Proposition I.8: If two triangles have the two sides equal to two sides respectively, and have also the base equal to the base, they will also have the angles equal which are contained by the equal straight lines.

Being what it was required to prove.