Commentary on Proposition IV.6¶
Back to Proposition IV.6.
Euclid here proceeds to consider problems corresponding to those in Props. 2—5 with reference to figures of four or more sides, but with the difference that, whereas he dealt with triangles of any form, he confines himself henceforth to regular figures. It happened to be as easy to divide a circle into three parts which are in the ratio of the angles, or of the supplements of the angles, of a triangle as into three equal parts. But, when it is required to inscribe in a circle a figure equiangular to a given quadrilateral, this can only be done provided that the quadrilateral has either pair of opposite angles equal to two right angles. Moreover, in this case, the problem may be solved in the same way as that of IV. 2, i.e. by simply inscribing a triangle equiangular to one of the triangles into which the quadrilateral is divided by either diagonal, and then drawing on the side corresponding to the diagonal as base another triangle equiangular to the other triangle contained in the quadrilateral. But this is not the only solution; there are an infinite number of other solutions in which the inscribed quadrilateral will, unlike that found by this particular method, not be of the same form as the given quadrilateral. For suppose ABCD to be the quadrilateral inscribed in the circle by the method of IV. 2. Take any point B' on AB, join AB', and then make the angle DAD' (measured towards AC) equal to the angle BAB'. Join B'C, CD'. Then AB'CD' is also equiangular to the given quadrilateral, but not of the same form. Hence the problem is indeterminate in the case of the general quadrilateral. It is equally so if the given quadrilateral is a rectangle; and it is determinate only when the given quadrilateral is a square.