EXERCISES IN INVENTION. THEOREMS. 1. In any circumscribed quadrilateral, the sum of two opposite sides is equal to the sum of the other two sides. 2. A quadrilateral is inscriptible if two of its opposite angles are supplements of each other. 3. The bisectors of the angles formed by producing the opposite sides of an inscribed quadrilateral intersect at right angles. 4. If a circle is described on the radius of another circle, any straight line drawn from the point of contact to the outer circumference is bisected by the interior one. PROBLEMS. 1. To trisect a right angle. 2. Given two lines that would meet if sufficiently produced, to find the bisector of their included angle without finding its vertex. 3. To draw a common tangent to two given circles. 4. Inscribe a square in a given rhombus. 5. To construct a square, given its diagonal. 6. Construct an angle of 30°, one of 60°, one of 120°, one of 150°, one of 45°, and one of 135°. 7. Construct a triangle, given the base, the angle opposite the base, and the medial line to the base. 8. Construct a triangle, given the vertical angle, and the radius of the circumscribing circle. 9. Construct a triangle, given the base, the vertical angle, and the perpendicular from the extremity of the base to the opposite side. 10. Construct a triangle, given the base, an angle at the base, and the sum or difference of the other two sides. 11. Construct a square, given the sum or difference of its diagonal and side. 12. Describe a circle cutting the sides of a square, so as to divide the circumference at the points of intersection into eight equal arcs. 13. Through any point within a circle, except the centre, to draw a chord which shall be bisected at that point. BOOK IV. AREA AND RELATION OF POLYGONS. DEFINITIONS. 237. Similar Polygons are polygons which are mutually equiangular, and have their homologous sides proportional. 238. The Area of a polygon is its quantity of surface; it is expressed by the number of times the polygon contains some other area taken as a unit of measure. The unit of measure usually assumed is a square, a side of which is some linear unit; as, a square inch, a square foot, etc. 239. Equivalent Figures are such as have equal areas. AREAS. THEOREM I. 240. The area of a rectangle equals the product of its base and altitude. Let ABCD be a rectangle, AB the base, and AC the altitude. Let AE be a common unit of measure of the sides AB and AC, and suppose it to be contained in AB 5 times, and in AC 3 times. Apply AE to AB and AC, dividing them respectively into five and three equal parts. Through the several points of division draws to the sides. The rectangle will then be divided into equal squares, as the angles are all Ls, and the sides all equal. (130) Now, the whole number of these squares is equal to the number in the row on AB multiplied by the number of rows, or the number of linear units in AB multiplied by the number in AC. Now, this is true, whatever may be the length of the common unit of measure; hence it is true if it is infinitely small, as is the case when the sides are incommensurable. Therefore, in any case, the proposition is true. Q.E. D. |