121. Construct a polygon of an odd number of sides, being given that the sides taken in order are divided in given ratios by fixed points. 122. If the external diagonal of a quadrilateral inscribed in a given circle be a chord of another given circle, the locus of its middle point is a circle. 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 _______ 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. PROP. I.—Theorem. One part (AB) of a right line cannot be in a plane (X), and another part (BC) not in it. Dem.—Since AB is in the plane X, it can be produced in it [Bk. I. Post. ii.]; let it be produced to D. Then, if BC be not in X, let any other plane passing through AD be turned round AD until it passes through the point C. Now, because the points B, C are in this second plane, the line BC [I., Def. vi.] is in it. Therefore the two right lines ABC, ABD lying in one plane have a common segment AB, which is impossible. Therefore, &c. PROP. II.—Theorem. Two right lines (AB, CD) which intersect one another in any point (E) are coplanar, and so also are any three right lines (EC, CB, BE) which form a triangle. Dem.—Let any plane pass through EB, and be turned round it until it passes through C. Then because the points E, C are in this plane, the right line EC is in it [I., Def. vi.]. For the same reason the line BC is in it. Therefore the lines EC, CB, BE are coplanar; but AB and CD are two of these lines. Hence AB and CD are coplanar. PROP. III.—Theorem. If two planes (AB, BC) cut one another, their common section (BD) is a right line. Dem.—If not from B to D, draw in the plane AB the right line BED, and in the plane BC the right line BFD. Then the right lines BED, BFD enclose a space, which [I., Axiom x.] is impossible. Therefore the common section BD of the two planes must be a right line. PROP. IV.—Theorem. If a right line (EF) be perpendicular to each of two intersecting lines (AB, CD), it will be perpendicular to any line GH, which is both coplanar and concurrent with them.