18. If the middle points of the three sides of a triangle be joined by straight lines, the four triangles formed will be equal in area. Draw CD parallel to AB and produce FE to D. Show now the equality of the triangles AEF, DEC. Hence EF is parallel to BC. Similarly, FG B is parallel to AC, and EG to AB. Hence the proposition may be deduced. A G 19. Given the middle points of the three sides of a triangle, to construct the triangle. 20. Given the middle points of the four sides of a parallelogram, to construct the parallelogram. 21. The square on a side of a triangle subtending an acute angle, is less than the squares on the other sides. Let BAC be an acute angle, then BC2<AC2+AB2. = Make AD AC, and at right angles to AB (I. 22). D A 22. The square on the side subtending the obtuse angle, is greater than the squares on the other sides. 23. Any side of a triangle is greater than the difference between the other two. 24. Two angles are equal if their sides be parallel, each to each, and lying in the same direction (I. 27), (Ax. 3). 25. To find the side of a square equal in area to two given squares (I. 42). 26. If from the ends of one side of a triangle straight lines be drawn to a point within the triangle, these lines shall contain a greater angle than the other two sides of the triangle (I. 18). 27. If each of the equal angles of an isosceles triangle be double the third angle, then the exterior angle formed by producing one of the equal sides beyond the base is three times the third angle. 28. In the last example how many degrees in the various angles of the figure? 29. If the equal sides of an isosceles triangle be produced beyond the base, the angles on the other side of the base will be equal. 30. If one angle of a parallelogram contain 40°, what is the value of each of the others? 31. If a line be drawn bisecting an angle, any point of it is equally distant from the sides of the angle. 32. The lines bisecting the three angles of a triangle all intersect in the same point. 33. One of the angles of a parallelogram is three halves of a right angle. What are the values of the others in parts of a right angle? in degrees? 34. One of the exterior angles of an equilateral figure is of a right angle. How many sides has the figure? 35. Construct a five-sided figure, four of whose sides are 3, 4, 5, 6, and whose angles in the same order are 2, 1, 1, 1⁄2, 1, right angles. BOOK II. RECTANGLES. DEFINITIONS. 1. Every right-angled parallelogram, or rectangle, is said to be contained by any two of the straight lines which contain one of the right angles. Thus, the rectangle AC is said to be contained by AB and BC, and is called the rectangle AB. BC, or simply AB. BC. When we speak of the rectangle of two disconnected straight lines, as A and B, we mean the right-angled parallelogram of which those lines are the adjacent sides. Thus, make EC equal to A, and CD, at right angles to it, equal to B, and complete the parallelogram. The rectangle A.B is ED. E A B B A D By rectangle we mean the figure formed, and not the product of the lines. We will show hereafter that the area of the rectangle is equal to the product of the number of units in the two sides. 2. In every parallelogram the figure formed by either of the parallelograms about the diameter, together with the two complements, is called a gnomon. Theorem. If there be two straight lines, one of which is divided into any number of parts, the rectangle contained by the two straight lines, is equal to the rectangles contained by the undivided line and the several parts of the divided line. Let A and BC be two straight lines; and let BC be divided into any number of parts at the points D, E; the rectangle contained by the straight lines A, BC, will be equal to the rectangle contained by A, BD, together with that contained by A, DE, and that contained by A, EC. From the point B draw (I. 11) BF at right angles to BC; and make (I. 1) BG equal to A; through G draw (I. 29) GH parallel to BC; and through D, E and C draw (I. 29) DK, EL and CH parallel to BG. Then the rectangle BH is equal to the rectangles BK, DL, EH. F B D E C But BH=BG: BC=A.BC, because A is equal to BG. Also, BK-BG.BD = A. BD; DL-DK.DE=A.DE; EHEL. EC=A.EC; A.BC- A. BD+A.DE+A. EC. H Proposition 2. Theorem. If a straight line be divided into any two parts, the rectangles contained by the whole and each of the parts, are together equal to the square of the whole line. Theorem.-If a straight line be divided into any two parts, the rectangle contained by the whole and one of the parts, is equal to the rectangle contained by the two parts, together with the square of the aforesaid part. Let the straight line AB be divided in two parts in the point C, then AB. BC= AC. CB+BC2. Upon BC describe (I. 41) the square CE; produce ED to F, and draw AF parallel to CD. A B E |