Ex. 10. AD is drawn perpendicular to BC the opposite side of a triangle ABC; prove that AB>BD and AC>CD. Hence show that AB+ AC>BC. [There will be two cases.] Ex. 11. The bisectors of the angles B, C of a triangle ABC intersect at O. Prove that, if Ab>AC, OB>OC. Ex. 12. If the perpendiculars from B, C to the opposite sides of the triangle ABC intersect at a point X inside the triangle, and if AB>AC, prove that XB> XC. Ex. 13. The sides AB, AC of a triangle are produced, and the bisectors of the external angles at B, C intersect at E. Prove that, if Ab> ac, eb<EC. Ex. 14. A straight line cuts the equal sides AB, AC of an isosceles triangle ABC in X, Y and cuts the base BC produced towards C. Prove that AY>AX. Ex. 15. Prove that the straight line joining the vertex of an isosceles triangle to any point in the base produced is greater than either of the equal sides. Ex. 16. Prove that the straight line joining the vertex of an isosceles triangle to any point in the base is less than either of the equal sides of the triangle. Ex. 17. If P is any point on the external bisector of the angle A of a triangle ABC, AB+ AC<PB+PC. Ex. 18. If D is any point in the side AC of a triangle ABC, prove that BA+AC > BD + DC. Ex. 19. If O is any point inside a triangle ABC, prove that BA+AC > BO+OC. [Produce BO to cut AC.] Ex. 20. O is a point inside a triangle ABC; prove that ▲ BOC> <BAC. [Produce BO to cut AC.] Ex. 21. Any three sides of a quadrilateral are together greater than the fourth side. Ex. 22. Two sides of a triangle are together greater than twice the median drawn through their point of intersection. [Produce the median AD to E, making DE=AD; join CE.] Ex. 23. O is a point inside a quadrilateral ABCD; prove that OA+OB+OC+OD cannot be less than AC + BD. Ex. 24. The sum of the distances of any point O from the vertices of a triangle ABC is greater than half the perimeter of the triangle. [The perimeter of a figure is the sum of its sides. Apply Th. 18 to as OBC, OCA, OAB in turn and add up the results.] Ex. 25. The sum of the distances from the vertices of a triangle of any point within the triangle is less than the perimeter of the triangle. [Apply Ex. 19 three times.] Would this be true for a point outside the triangle? Ex. 26. The sum of the diagonals of a quadrilateral is greater than half its perimeter. Ex. 27. The sum of the diagonals of a quadrilateral is less than its peri meter. Ex. 28. A and B are two points on the same side of a straight line CD; find the point P in CD for which AP+PB is least. Give a proof. [Join A to the image of B.] Ex. 29. ABC, APQC are a triangle and a convex quadrilateral on the same base AC, P and Q being inside the triangle; prove that the perimeter of the triangle is greater than that of the quadrilateral. [Produce AP, PQ to meet BC and use Th. 18.] Ex. 30. If ABC be a triangle and BPQR...C any convex polygon having BC for one side, and its other vertices P, Q, R, etc. lying inside ABC, show that the perimeter of ABC is greater than the perimeter of BPQR...C. A Ex. 31. Any chord of a circle which does not pass through the centre is less than a diameter. [Join the ends of the chord to the centre.] B Ex. 32. In fig. 38, O is the centre of the circle and POA a straight line; prove that PA> PB. Ex. 33. In fig. 38 prove that PC<PB. с Fig. 38. CHAPTER VI THE PARALLELOGRAM NOTE ON DEFINITIONS. Many definitions are mere explanations of terms and have no logical importance; e.g. median, chord. Definitions of this description are not included in the text, but collected in a table at the end of the book. In other cases, a figure is defined by selecting certain of its properties, and it is necessary to prove the existence of other properties. The best example of this is the parallelogram. DEF. A quadrilateral with its opposite sides parallel is called a parallelogram. You are familiar with other properties of a parallelogram, e.g. that opposite sides are equal; these are not enumerated in the definition, for a definition should not contain anything superfluous or redundant; these further properties can be deduced from the above sufficient definition. It would be quite reasonable to select another set of properties of a parallelogram for purposes of definition; e.g. we might have defined a parallelogram as a quadrilateral with one pair of opposite sides equal and parallel; or as a plane quadrilateral with opposite sides equal. But for a logical sequence it is necessary to choose one definition and adhere to it. A quadrilateral is not necessarily a plane figure; if a plane quadrilateral of paper is bent about a diagonal, a quadrilateral is obtained which is not plane; it is called a skew quadrilateral. However, in books on plane geometry the term quadrilateral is for convenience confined to plane quadrilaterals. But it should be noticed that the above definition is adequate even if "quadrilateral" is not defined as a plane figure. Two parallel straight lines define a plane; we have no need to limit "quadrilateral" to "plane quadrilateral" in this case, for the word "parallel" ensures that the figure shall be plane. THEOREM 20. THE PARALLELOGRAM. (i) The opposite sides of a parallelogram are equal. (ii) The opposite angles of a parallelogram are equal. (iii) Each diagonal bisects the parallelogram. Data ABCD is a parallelogram, and BD Simly BD (by joining AC)†. And since ▲ ABD = ▲ CDB, BD bisects the parallelogram. + This fact also follows by adding the two lines marked Th. 5. Th. 5. Th. 5. (iv) The diagonals of a parallelogram bisect one another. Fig. 40. OA = OC and OD = OB. Data ABCD is a parallelogram; its diagonals AC, BD intersect at O. [AS OAD, OCB must be proved congruent.] Th. 5. COR. 1. If a parallelogram has one of its angles a right angle, all its angles must be right angles. COR. 2. If one pair of adjacent sides of a parallelogram are equal, all its sides are equal. THEOREM 21. [CONVERSES OF THEOREM 20.] (i) A quadrilateral is a parallelogram if both pairs of opposite angles are equal. Data ABCD is a quadrilateral in which LA=LC = L x (say) and ▲ B = LD = Ly (say). To prove that ABCD is a parallelogram. |