Connections

Isn't it interesting how the solution of a simply-stated but famous problem may immediately resolve another such problem? Connections between the problems may be subtle and insightful-but they may also be quite glaring.

Pitfalls in such logical or demonstrative manipulations are wellknown, one of the most obvious is circularity. One would hardly use the law of cosines to prove the Theorem of Pythagoras had the famous theorem first been used to derive the cosine law. Notice how the distance formula, which is a disguised form of the Theorem of Pythagoras, permits the assertion of the law of cosines.

Likewise, one should not endeavor to prove Euclid's fifth postulate by using any one of its classical equivalents (say that of Bolyai, Proclus, Gauss, Legendre or Lobachevski). Perhaps the assertion by John Bolyai (1802-1860) is best known: namely, the familiar relationship that the angle sum of a triangle is 180 degrees.