123. If a chord of one circle be a tangent to another, the line connecting the middle point of each arc which it cuts off on the first, to its point of contact with the second, passes through a given point.
124. From a point P in the plane of a given polygon perpendiculars are let fall on its sides; if the area of the polygon formed by joining the feet of the perpendiculars be given, the locus of P is a circle.
BOOK XI.
THEORY OF PLANES, COPLANAR LINES, AND SOLID ANGLES
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DEFINITIONS.
i. When two or more lines are in one plane they are said to be coplanar.
ii. The angle which one plane makes with another is called a dihedral angle.
iii. A solid angle is that which is made by more than two plane angles, in different planes, meeting in a point.
iv. The point is called the vertex of the solid angle.
v. If a solid angle be composed of three plane angles it is called a trihedral angle; if of four, a tetrahedral angle; and if of more than four, a polyhedral angle.