cumference in the points A and E. Draw A B and E B, and each will be a tangent as required. For, drawing CA, the angle CA B, being inscribed in a semicircle, is a right angle (Theo. X. Cor. 2, Bk. III.); therefore A B is perpendicular to the radius CA at its extremity, A, and consequently is a tangent (Theo. VII. Bk. III.). In like manner it may be shown that E B is a tangent. EXERCISES FOR ORIGINAL THOUGHT, ON REVIEW. 170. 1. The diameter is the longest straight line that can be inscribed in a circle. 2. A straight line cannot meet the circumference of a circle in more than two points. 3. Two parallel tangents meet the circumference at the extremity of the same diameter. C DI B 4. If two straight lines intercept equal arcs of a circle, and do not cut each other within the circle, the lines will be parallel. 5. If a straight line be drawn to touch a circle, and be parallel to a chord, the point of contact will be the middle point of the arc cut off by that chord. 6. If two circumferences cut each other, the straight line passing through their centers will bisect at right angles the chord which joins the points of intersection. 7. At a point in a given straight line to make an angle equal to half a right angle. 8. To describe a square whose side shall be equal to a given straight line. BOOK IV. AREAS AND RELATIONS OF PLANE FIGURES. DEFINITIONS. 171. The Area of a figure is its quantity of surface, and is expressed by the number of times which the surface contains some other area assumed as a unit of measure. Figures have equal areas, when they contain the same unit of measure an equal number of times. 172. Similar Figures are such as have the angles of the one equal to those of the other, each to each, and the sides containing the equal angles proportional. 173. Equivalent Figures are such as have equal areas. Figures may be equivalent which are not similar. Thus a circle may be equivalent to a square, and a triangle to a rectangle. 174. Equal Figures are such as, when applied the one to the other, coincide throughout (26, Ax. 12). Thus circles having equal radii are equal; and triangles having the three sides of the one equal to the three sides of the other, each to each, are also equal. Equal figures are always similar; but similar figures may be very unequal. 175. In different circles, Similar Arcs, Segments, or Sectors are such as corre E B D H to the segment FHG, and the sector A B C to the sector EFG. 176. The Altitude of a Triangle is the perpendicular, which measures the distance of any one of its vertices from the opposite side taken as a base; as the perpendicular A D let fall on the base BC in the triangle ABC. A B D E D C F B 177. The Altitude of a Parallelogram is the perpendicular which measures the distance between its opposite sides taken as bases; as the perpendicular E F measuring the distance between A the opposite sides, A B, DC, of the parallelogram A B C D. 178. The Altitude of a Trapezoid is the perpendicular distance between its parallel sides; as the distance measured by the perpendicular E F between the parallel sides, A B, D C, of the trapezoid ABCD. E D C A F B THEOREM I. 179. Parallelograms which have equal bases and equal altitudes are equivalent. Let A B CD, ABEF be two parallelograms having equal bases and equal altitudes; then these parallelograms are equivalent. D C F E A B Let the base of the one parallelogram be placed on that of the other, so that A B shall be the common base. Now, since the two parallelograms are of the same altitude, their upper bases, DC, FE, will be in the same straight line, D C E F, parallel to A B. From the na ture of parallelograms DC is equal to A B, and FE is equal to AB (Theo. XXII. Bk. I.); therefore DC is equal to FE (26, Ax. 1); hence if DC and FE be taken away from the same line, DE, the remainders CE and D F will be equal (26, Ax. 3). But AD is equal to BC and AF to BE (Theo. XXII. Bk. I.); therefore the triangles DAF, CBE, are mutually equilateral, and consequently equal (Theo. XIII. Bk. I.). If from the quadrilateral A B E D, we take away the triangle A D F, there will remain the parallelogram A B E F ; and if from the same quadrilateral A BED, we take away the triangle C B E, there will remain the parallelogram ABCD. Hence the parallelograms A B CD, ABEF, which have equal bases and equal altitude, are equivalent. 180. Cor. Any parallelogram is equivalent to a rectangle having the same base and altitude. THEOREM II. 181. If a triangle and a parallelogram have the same base and altitude, the triangle is equivalent to half the parallelogram. Let ABE be a triangle, and D ABCD a parallelogram having the same base, A B, and the same altitude; then will the triangle be equivalent to half the parallelo gram. C F E A B Draw A F, FE so as to form the parallelogram A B E F. Then the parallelograms A B C D, A BE F', having the same base and altitude, are equivalent (Theo. I.). But the triangle ABE is half the parallelogram ABEF (Theo. XXII. Cor. 1, Bk. I.); hence the triangle A B E is equivalent to half the parallelogram A B C D (26, Ax. 7). 182. Cor. 1. Any triangle is equivalent to half a rectangle having the same base and altitude, or to a rectangle either |