The Teaching of Geometry

by David Eugene Smith

For example, produce AC to X,
making
CX = CB.
Then ∠X = ∠XBC.
∴ ∠XBA > ∠X.
∴ AX > AB.
∴ AC + CB > AB.

The above proof is due to Euclid. Heron of Alexandria (first century A.D.) is said by Proclus to have given the following:

Let CX bisect ∠C.
Then ∠BXC > ∠ACX.
∴∠BXC > ∠XCB.
∴CB > XB.
Similarly, AC > AX.
Adding, AC + CB > AB.

Theorem. If two sides of a triangle are unequal, the angles opposite these sides are unequal, and the angle opposite the greater side is the greater.

Euclid stated this more briefly by saying, "In any triangle the greater side subtends the greater angle." This is not so satisfactory, for there may be no greater side.

Theorem. If two angles of a triangle are unequal, the sides opposite these angles are unequal, and the side opposite the greater angle is the greater.

Euclid also stated this more briefly, but less satisfactorily, thus, "In any triangle the greater angle is subtended by the greater side." Students should have their attention called to the fact that these two theorems are reciprocal or dual theorems, the words "sides" and "angles" of the one corresponding to the words "angles" and "sides" respectively of the other.

It may also be noticed that the proof of this proposition involves what is known as the Law of Converse; for

(1) if b = c, then ∠B = ∠C;
(2) if b > c, then ∠B > ∠C;
(3) if b < c, then ∠B < ∠C;