The Teaching of Geometry

by David Eugene Smith

It is interesting to a class to have a teacher point out that, in this figure, AP + PB < AC + CB, and AP' + P'B < AP + PB, and that the nearer P gets to AB, the shorter AP + PB becomes, the limit being the line AB. From this we may infer (although we have not proved) that "a straight line (AB) is the shortest path between two points."

Theorem. Only one perpendicular can be drawn to a given line from a given external point.

Theorem. Two lines drawn from a point in a perpendicular to a given line, cutting off on the given line equal segments from the foot of the perpendicular, are equal and make equal angles with the perpendicular.

Theorem. Of two lines drawn from the same point in a perpendicular to a given line, cutting off on the line unequal segments from the foot of the perpendicular, the more remote is the greater.

Theorem. The perpendicular is the shortest line that can be drawn to a straight line from a given external point.

These four propositions, while known to the ancients and incidentally used, are not explicitly stated by Euclid. The reason seems to be that he interspersed his problems with his theorems, and in his Propositions 11 and 12, which treat of drawing a perpendicular to a line, the essential features of these theorems are proved. Further mention will be made of them when we come to consider the problems in question. Many textbook writers put the second and third of the four before the first, forgetting that the first is assumed in the other two, and hence should precede them.

Theorem. Two right triangles are congruent if the hypotenuse and a side of the one are equal respectively to the hypotenuse and a side of the other.

Theorem. Two right triangles are congruent if the hypotenuse and an adjacent angle of the one are equal respectively to the hypotenuse and an adjacent angle of the other.

As stated in the notes on the third proposition in this sequence, Euclid's cumbersome Proposition 26 covers several cases, and these two among them. Of course this present proposition could more easily be proved after the one concerning the sum of the angles of a triangle, but the proof is so simple that it is better to leave the proposition here in connection with others concerning triangles.