Substituting these values, we have 2A+2B = 4R, or, A+B = 2R: AD || BC (11, 3). In like manner it may be proved that AB || DC. XLI. Theorem. The diagonals of a parallelogram bisect each other. HYPOTH. AC and BD are diagonals of A the parallelogram ABCD. TO BE PROVED. AE= EC, and BE = ED. PROOF. In the triangles AEB and CED, B E COR. 1. If AB = BC, then ▲ AEB CEB, being mutually equi- hence AEB = BEC = R; or, BE AC (3, 4). Hence the diagonals of a rhombus bisect each other at right angles. A B COR. 2. If the parallelogram ABCD is right angled, ▲ ADC BCD: hence AC = BD; or, the diagonals of a rectangle are equal. XLII. Theorem. If a series of parallels cutting any two straight lines intercept equal distances on one of these lines, they also intercept equal distances on the other line. HYPOTH. The two lines AB and CD are cut by the paralintercepting on CD the dis lels, aa', bb', cc', dd', tances ab = bc cd= = A COR. 1. If in the triangle ABC, AD = DB, and DE || BC, then AE EC. Also since two points, D and E, fix the position of a straight line, it follows, that if AD = DB, and AE = EC, then DE || BC ; that is, the line joining the middle points of two sides of a triangle is parallel to the third side. COR. 2. Since DE BF FC, DE = = BC B D that is, the line joining the middle points of two sides of a triangle is equal to one-half the third side. COR. 3. If, in the trapezoid ABCD, E and F are the middle points of the non-parallel sides, it follows that EF || AB. Also draw the diagonal DB; then, by Cor. 2, That is, the line joining the middle points of the nonparallel sides of a trapezoid is parallel to the other two sides, and is equal to one-half their sum. EXERCISES. 1. In an oblique-angled parallelogram, the diagonal joining the vertices of the smaller angles is the greater. 2. The diagonals of a rhombus bisect its angles. 3. The lines joining the middle points of the sides of a triangle divide it into four congruent triangles. 4. The triangle formed by lines drawn through the vertices of the angles of any triangle parallel to the opposite sides is divided into four congruent triangles. 5. If the line bisecting an angle of a triangle bisects also the opposite side, it is perpendicular to that side, and the triangle is isosceles. A Prolong the bisecting line AD until DE=AD; join, &c. 6. If the base of an isosceles triangle be produced, the exterior angle, minus half the vertical angle, is equal to one right angle. 7. If BC is the base of an isosceles triangle, ABC, and BD is drawn perpendicular to AC, the angle DBC=BAC. 8. The base BC of an isosceles triangle, ABC, is parallel to a line bisecting the exterior angle at A. 9. If, from a variable point in the base of an isosceles triangle, lines be drawn parallel to the equal sides, the perimeter of the parallelogram thus formed is constant. 10. Any line bisecting the diagonal of a parallelogram bisects the parallelogram. BOOK II. THE CIRCLE. I. DEFINITIONS. 1. A circle is a portion of a plane bounded by a curved line every point of which is equally distant from a point within called the centre. A F E B 2. The curve is called the circumference of the circle. 3. An arc is a portion of the circumference; as, AD. 4. A radius is a line drawn from the centre to the circumference; as, OC or OA. 5. A diameter is a line passing through the centre, and terminated each way by the circumference; as, AB. COR. 1. A radius is one-half a diameter: all radii of the same circle are equal; and they measure the distance of the centre from the circumference. COR. 2. A point at a greater distance from the centre is without the circle, and a point at a less distance is within the circle. Hence the circumference is the locus of all the points of a plane that are equally distant from a given point. COR. 3. Circles with equal radii are congruent; for, if we apply the centres to each other, the circumferences will coin 34 cide: otherwise some points would be at a greater or less distance from the centre than the radius. 6. A chord is a line joining any two points of the circumference; as, DE. 7. A segment is the part of the circle included between an arc and its chord; as, DFE. 8. A sector is the figure included between an arc and the radii drawn to its extremities; as, BOC. II. Theorem. A diameter divides the circle and its circumference into two congruent parts. HYPOTH. AB is a diameter of the DEF. The segment ACB is called a semicircle; and the arc ACB, a semi-circumference. III. Theorem. A straight line cannot have more than two Any point, F, between A and B, is nearer the centre, C |