ADDITIONAL EXERCISES. BOOK I. 1. Every point within an angle, and not in the bisector, is unequally distant from the sides of the angle. (Prove by Reductio ad Absurdum.) 2. If two lines are cut by a third, and the sum of the interior angles on the same side of the transversal is less than two right angles, the lines will meet if sufficiently produced. (Prove by Reductio ad Absurdum.) 3. State and prove the converse of Prop. XXXVII., (Prove BAD + 2B = 180°.) 4. The bisectors of the exterior angles of a triangle form a triangle whose angles are respectively the half-sums of the angles of the given triangle taken two and two. (Ex. 69, p. 67.) (To prove ▲A' = } (Z ABC + ≤BCA), etc.) II. B B' 5. If CD is the perpendicular from C to side AB of triangle ABC, and CE the bisector of angle C, prove DCE equal to onehalf the difference of angles A and B. 6. If E, F, G, and H are the middle points of sides AB, BC, CD, and DA, respectively, of quadrilateral ABCD, prove EFGH a parallelogram whose perimeter is equal to the sum of the diagonals of the quadrilateral. (§ 130.) 7. The lines joining the middle points of the opposite sides of a quadrilateral bisect each other. (Ex. 6, p. 220.) 8. The lines joining the middle points of the opposite sides of a quadrilateral bisect the line joining the middle points of the diagonals. (EKGL is a, and its diagonals bisect each other.) B F H 9. The line joining the middle points of the diagonals of a trapezoid is parallel to the bases and equal to one-half their difference. B 10. If D is any point in side AC of triangle ABC, and E, F, G, and H the middle points of AD, CD, BC, and AB, respectively, prove EFGH a parallelogram. 11. If E and G are the middle points of sides AB and CD, respectively, of quadrilateral ABCD, and K and L the middle points of diagonals AC and BD, respectively, prove ▲ EKL = ▲ GKL. 12. If D and E are the middle points of sides BC and AC, respectively, of triangle ABC, and AD be produced to F and BE to G, making DF = AD and EG = BE, prove that line FG passes through C, and is bisected at that point. 13. If D is the middle point of side BC of triangle ABC, prove AD< } (AB + AC). (Produce AD to E, making DE = AD.) 14. The sum of the medians of a triangle is less than the perimeter, and greater than the semi-perimeter of the triangle. (Ex. 13, p. 221, and Ex. 106, p. 71.) 15. If the bisectors of the interior angle at C and the exterior angle at B of triangle ABC meet at D, prove BDC = } ≤ A. 16. If AD and BD are the bisectors of the exterior angles at the extremities of the hypotenuse of right triangle ABC, and DE and DF are drawn perpendicular, respectively, to CA and CB produced, prove CEDF a square. (D is equally distant from AC and BC.) 17. AD and BE are drawn from two of the vertices of triangle ABC to the opposite sides, making ▲ BAD = ▲ ABE; if AD = BE, prove the triangle isosceles. 18. If perpendiculars AE, BF, CG, and DH, be drawn from the vertices of parallelogram ABCD to any line in its plane, not intersecting its surface, prove (The sum of the bases of a trapezoid is equal to twice the line joining the middle points of the non-parallel sides.) 19. If CD is the bisector of angle C of triangle ABC, and DF be drawn parallel to AC meeting BC at E and the bisector of the angle exterior to Cat F, prove DE = EF. D C B E 20. If E and F are the middle points of sides AB and AC, respectively, of triangle ABC, and AD the perpendicular from A to BC, prove / EDF = LEAF. (Ex. 83, p. 69.) 21. If the median drawn from any vertex of a triangle is greater than, equal to, or less than one-half the opposite side, the angle at that vertex is acute, right, or obtuse, respectively. (§ 98.) 22. The number of diagonals of a polygon of n sides is n (n − 3). 2 23. The sum of the medians of a triangle is greater than threefourths the perimeter of the triangle. (Fig. of Prop. LII. Since A0 = } AD and B0 = } BE, we have AB< (AD + BE), by Ax. 4.) 24. If the lower base AD of trapezoid ABCD is double the upper base BC, and the diagonals intersect at E, prove CE = { AC and BE = BD. (Let F be the middle point of DE, and G of AE.) 25. If O is the point of intersection of the medians AD and BE of equilateral triangle ABC, and line OF be drawn parallel to side AC, meeting side BC at F, prove that DF is equal to † BC. (§ 133.) (Let G be the middle point of OA.) A B B G F C DFH 26. If equiangular triangles be constructed on the sides of a triangle, the lines drawn from their outer vertices to the opposite vertices of the triangle are equal. (§ 63.) 27. If two of the medians of a triangle are equal, the triangle is isosceles. (Fig. of Prop. LII. Let AD = BE.) BOOK II. 28. AB and AC are the tangents to a circle from point A, and D is any point in the smaller of the arcs subtended by chord BC. If a tangent to the circle at D meets AB at E and AC at F, prove the perimeter of triangle AEF constant. (§ 174.) 29. The line joining the middle points of the arcs subtended by sides AB and AC of an inscribed triangle ABC cuts AB at F and AC at G. Prove AF- AG. (LAFG = LAGF.) 30. If ABCD is a circumscribed quadrilateral, prove the angle between the lines joining the opposite points of contact equal to (4+ C). (§ 202.) 31. If sides AB and BC of inscribed hexagon ABCDEF are parallel to sides DE and EF, respectively, prove side AF parallel to side CD. (§ 172.) (Draw line CF, and prove ▲ AFC = 2 FCD.) 32. If AB is the common chord of two intersecting circles, and AC and AD diameters drawn from A, prove that line CD passes through B. (§ 195.) D 33. If AB is a common exterior tangent to two circles which touch each other externally at C, prove ACB a right angle. (Draw the common tangent at C, meeting AB at D.)` 34. If AB and AC are the tangents to a circle from point A, and D is any point on the greater of the arcs subtended by chord BC, prove the sum of angles ABD and ACD constant. 35. If A, C, B, and D are four points in a straight line, B being between C and D, and EF is a common tangent to the circles described upon AB and CD as diameters, prove 37. ABCD is a quadrilateral inscribed in a circle. If sides AB and DC produced intersect at E, and sides AD and BC produced H at F, prove the bisectors of angles E and F perpendicular. (§ 199.) (Prove arc HM + arc KL = = 180°.) E A B O' 38. If ABCD is an inscribed quadrilateral, and sides AD and BC produced meet at P, the tangent at P to the circle circumscribed about triangle ABP is parallel to CD. (§ 196.) (Prove between the tangent and BP equal to ≤ PCD.) 39. ABCD is a quadrilateral inscribed in a circle. Another circle is described upon AD as a chord, meeting AB and CD at E and F, respectively. Prove chords BC and EF parallel. 40. If ABCDEFGH is an inscribed octagon, the sum of angles A, C, E, and G is equal to six right angles. (§ 193.) 41. If the number of sides of an inscribed polygon is even, the sum of the alternate angles is equal to as many right angles as the polygon has sides less two. (Use same method of proof as in Ex. 40.) 42. If a right triangle has for its hypotenuse the side of a square, and lies without the square, the straight line drawn from the centre of the square to the vertex of the right angle bisects the right angle. (§ 200.) 43. The perpendiculars from the vertices of a triangle to the opposite sides are the bisectors of the angles of the triangle formed by joining the feet of the perpendiculars. (To prove AD, BE, and CF the bisectors of the of ▲ DEF. By § 200, a O can be circumscribed about quadrilateral BDOF; then Z ODF ZOBF; in this way, ODF 90° - Z BAC.) CONSTRUCTIONS. B E 44. Given a side, an adjacent angle, and the radius of the circumscribed circle of a triangle, to construct the triangle. What restriction is there on the values of the given lines? 45. To describe a circle of given radius tangent to a given circle, and passing through a given point without the circle. 46. To draw between two given intersecting lines a straight line which shall be equal to one given straight line, and parallel to another. (Draw a to one of the intersecting lines.) |