Commentary on Proposition III.26¶
Back to Proposition III.26.
As in III. 21, if Euclid’s proof is to cover all cases, it requires us to take cognisance of “angles at the centre” which are equal to or greater than two right angles. Otherwise we must deal separately with the cases where the angle at the circumference is equal to or greater than a right angle. The case of an obtuse angle at the circumference can of course be reduced by means of III. 22 to the case of an acute angle at the circumference; and, in case the angle at the circumference is right, it is readily proved, by drawing the radii to the vertex of the angle and to the other extremities of the lines containing it, that the latter two radii are in a straight line, whence they make equal bases in the two circles as in Euclid’s proof.
Lardner has another way of dealing with the right angle or obtuse angle at the circumference. In either case, he says, “bisect them, and the halves of them are equal, and it can be proved, as above, that the arcs upon which these halves stand are equal, whence it follows that the arcs on which the given angles stand are equal.”