Corollary. When the center of the circle falls within the triangle, the triangle is acute-angled; when the center falls on a side, the triangle is right-angled; and when the center of the circle falls outside the triangle, the triangle is obtuse-angled.
Corollary.
The side of the hexagon equals the radius of the circle.
And, in like manner as in the case of the pentagon, if through the points of division on the circle we draw tangents to the circle, there will be circumscribed about the circle an equilateral and equiangular hexagon in conformity with what was explained in the case of the pentagon.
And further by means similar to those explained in the case of the pentagon we can both inscribe a circle in a given hexagon and circumscribe one about it.
Corollary. And, in like manner as in the case of the pentagon, if through the points of division on the circle we draw tangents to the circle, there will be circumscribed about the circle a fifteen-angled figure which is equilateral and equiangular.
And further, by proofs similar to those in the case of the pentagon, we can both inscribe a circle in the given fifteen-angled figure and circumscribe one about it.
Figure | Inscribe figure in circle | Circumscribe figure about circle | Inscribe circle in figure | Circumscribe circle about figure |
Triangle | IV.2 | IV.3 | IV.4 | IV.5 |
Square | IV.6 | IV.7 | IV.8 | IV.9 |
Regular pentagon | IV.11 | IV.12 | IV.13 | IV.14 |
Regular hexagon | IV.15 | IV.15,Cor | IV.15,Cor | IV.15,Cor |
Regular 15-gon | IV.16 | IV.16,Cor | IV.16,Cor | IV.16,Cor |
There are only two other propositions. Proposition IV.1 is a basic construction to fit a line in a circle, and proposition IV.10 constructs a particular triangle needed in the construction of a regular pentagon.
Most of the propositions of Book IV are logically independent of each other. There is a short chain of deductions, however, involving the construction of regular pentagons.
Dependencies within Book IV | |
1, 5 | 10 |
2, 10 | 11 |
11 | 12 |
1, 2, 11 | 16 |