The Teaching of Geometry

by David Eugene Smith

Therefore C' will fall on both AC and BC, and hence at their intersection.

∴B'C' = AC.
But B'C' was made equal to BC.
∴AC = BC. Q.E.D.

If the proposition should be postponed until after the one on the sum of the angles of a triangle, the proof would be simpler, but it is advantageous to couple it with its immediate predecessor. This simpler proof consists in bisecting the vertical angle, and then proving the two triangles congruent. Among the other proofs is that of the reductio ad absurdum, which the student might now meet, but which may better be postponed. The phrase reductio ad absurdum seems likely to continue in spite of the efforts to find another one that is simpler. Such a proof is also called an indirect proof, but this term is not altogether satisfactory. Probably both names should be used, the Latin to explain the nature of the English. The Latin name is merely a translation of one of several Greek names used by Aristotle, a second being in English "proof by the impossible," and a third being "proof leading to the impossible." If teachers desire to introduce this form of proof here, it must be borne in mind that only one supposition can be made if such a proof is to be valid, for if two are made, then an absurd conclusion simply shows that either or both must be false, but we do not know which is false, or if only one is false.

Theorem. Two triangles are congruent if the three sides of the one are equal respectively to the three sides of the other.

It would be desirable to place this after the fourth proposition mentioned in this list if it could be done, so as to get the triangles in a group, but we need the fourth one for proving this, so that the arrangement cannot be made, at least with this method of proof.

This proposition is a "partial converse" of the second proposition in this list; for if the triangles are ABC and A'B'C', with sides a, b, c and a', b', c', then the second proposition asserts that if b = b', c = c', and ∠A = ∠A', then a = a' and the triangles are congruent, while this proposition asserts that if a = a', b = b', and c = c', then ∠A = ∠A' and the triangles are congruent.

The proposition was known at least as early as Aristotle's time. Euclid proved it by inserting a preliminary proposition to the effect that it is impossible to have on the same base AB and the same side of it two different triangles ABC and ABC', with AC = AC', and BC = BC'. The proof ordinarily given to-day, wherein the two triangles are constructed on opposite sides of the base, is due to Philo of Byzantium, who lived after Euclid's time but before the Christian era, and it is also given by Proclus. There are really three cases, if one wishes to be overparticular, corresponding to the three pairs of equal sides. But if we are allowed to take the longest side for the common base, only one case need be considered.

Of the applications of the proposition one of the most important relates to making a figure rigid by means of diagonals. For example, how many diagonals must be drawn in order to make a quadrilateral rigid? to make a pentagon rigid? a hexagon? a polygon of n sides. In particular, the following questions may be asked of a class:

1. Three iron rods are hinged at the extremities, as shown in this figure. Is the figure rigid? Why?