1. After showing people the two proofs of the Pythagorean Theorem, in Theorem 3.4, and after explaining to them that since the distance formula only gives us the Pythagorean Theorem in the case where the perpendicular sides of the triangle are horizontal and vertical, they would often ask, "Can't you just rotate the triangle until the sides are horizontal and vertical?" It turns out that the answer is "Yes.", but one needs to exercise a little caution.

Rotations are often defined using angles which are not developed until section 5, and one needs to check that the Pythagorean Theorem (Theorem 3.4) has not been used to obtain the results from that section which we would need to discuss rotations. We prove that a rotation is an isometry by showing that it is the composition of two reflections. The synthetic proof that a reflection preserves distances uses the Pythagorean theorem. This is not fatal since we also have the analytic proof, which uses only the distance formula; however, the fact that Theorem 5.1, from which everything in section 5 depends, uses the Pythagorean theorem (Theorem 3.4) is fatal.

It is possible to get around this problem by using an analytic proof which does not mention angles. It is possible to come up with the formulas for the coordinates of the image of a point after the plane has been rotated about the origin. The derivation of the formula uses trigonometry and linear algebra, both of which are beyond the scope of this development. That difficulty can be circumvented by simply producing the formula, springing full blown like Athena from the forehead of Zeus, without proving that it is the formula for finding the image of a point after the plane has been rotated about the origin. The map



where   m   is the slope of the line making the angle with the   x-axis, is the map which will rotate a plane so that a line whose slope is   m   will become horizontal. However, we do not need to prove that this map is a rotation. We need only prove the following.

Lemma 1:   R   is an isometry

Lemma 2:   R   will move a line whose slope is   m   to a horizontal line.

Lemma 3:   R   will move a line which is perpendicular to a line whose slope is   m   to a vertical line.

If the perpendicular sides in the image are horizontal and vertical, as follows from Lemmas 2 and 3, then the sides in the image satisfy the Pythagorean Theorem (Theorem 3.4) by the distance formula. If the map preserves distances, as follows from Lemma 1, then the lengths of the sides of the original triangle are the same as the lengths of the sides of the image of the triangle, and the sides in the original triangle, then, also satisfy the Pythagorean Theorem (Theorem 3.4).

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Analytic Foundations of Geometry

R. S. Wilson