TRACT 13.

SPHERE AND CONE.

24!)

Corol .—The great circles passing through all the lines orchords which intersect in the point p, will all intersect in thecommon diameter dpf.

prop. 6.

If a sphere be placed within a cone, so as to touch it in twopoints; then shall the outside of the sphere, and the insideof the cone, mutually touch quite around, and the line ofcontact will be a circle.

Demons. —Let v be the vertex of thecone, c the centre of the sphere, t one ofthe two points of contact, and tv a sideof the cone. Draw ct, cv. Then tvcis a triangle right-angled at t (Prop. 2).

In like manner, t being another point ofcontact, and ct being drawn, the trianglefvc will be right-angled at t. These twotriangles then, tvc, tvc, having the two sides ct, tv, equalto the two ct, tv (Prop. 1), and the included angle t equalto the included angle f, will be equal in all respects (Lucki. 4), and consequently have the angle tvc equal to the angletvc.

Again, let fall the perpendiculars tp, tv. Then the twotriangles tvp, tvv, having the two angles tvp and tpv equal, to the two tv p and tpv, and the side tv equal to the side tv(Prop. 1), will be equal in all respects (Euci. i. 26); conse-quently tp is equal to tp, and vp equal to vp. Hence pt, p tare radii of a little circle of the sphere, whose^ plane is per-pendicuhrr to the line cv, and its circumference every whereequidistant from the point c or v. This circle is therefore acircular section both of the sphere and of the cone, and istherefore the line of their mutual contact. Also cv is the axisof the cone.

Corol. 1.—The axis of a cone, when produced, passesthrough the centre of the inscribed sphere.

Corol. 2.—Hence also, every cone circumscribing a sphere,so that their surfaces touch quite around, is a right cone;