16. Describe a circle which shall pass through a given point and shall touch a given straight line and a given circle. 17. Describe a circle which shall pass through a given point and be tangent to two given circles. 18. Describe a circle which shall be tangent to two given straight lines and to a given circle. 19. Draw a chord through a given point in the interior of a circle which shall be divided in a given ratio at the given point. 20. Through a given point exterior to a circle draw a secant such that the exterior part shall be four-fifths of the whole secant. 21. Construct a triangle, given the base, the line which joins the vertex to the middle point of base, and the ratio of the two other sides. 22. Construct a triangle, given two sides and the bisectrix of the angle which they contain. 23. Construct a triangle similar to a given triangle and the vertices of which shall rest on three given concentric circles. 24. Inscribe a square in a given triangle. 25. Find a point in the interior of a triangle so that the lines drawn from it to the three vertices of the triangle shall divide it into three equal triangles. 26. Find the radical axis of two circles which do not intersect or touch one another. 27. Three circumferences being given, find upon one of them a point so that the tangents drawn from this point to the other two circumferences shall be equal. 28. Three circles being given, find on one of them a point so that the difference of the squares of the tangents drawn from this point to the two other circumferences shall be equal to a given square. 29. Given two points and a circumference, find on this circumference a point, C, the distances of which from the two points, A and B, shall be in the ratio of two given lines. 30. Given a circumference and a triangle, ABC, find on the circumference a point, X, so that the sum of the squares of the distances of this point from the three vertices, A, B, C, shall be equivalent to a given square. NUMERICAL PROBLEMS. 1. Find the area of a triangle whose base is 548 yards and perpendicular 265 meters. 2. Find the area of a trapezoid whose two parallel sides are 48. 2 meters, 30.5 meters, and altitude 27.45 meters. (Compute the same in feet, the meter being 39.37 inches, nearly.) 3. Compute the area of a parallelogram whose base is 145.6 meters, and altitude 72.48 meters. 4. Two sides of a triangle are respectively 15 and 12 feet, and the altitude corresponding to the third side is 9 feet. Find the third side and the distances of the other two sides from the opposite vertices. 5. The sides about the right angle of a R. A. T. are 3 meters and 4 meters respectively. Determine, to within a centimeter, First.—The hypothenuse and corresponding altitude. Fourth.—The portions of the sides fixed by the points of contact of the inscribed circles. Fifth.—The segments of each side determined by the bisectrix of the opposite angle. Sixth.—The lengths of the three bisectrices. 6. Two circles whose radii are respectively 1.2 meters and 3 decimeters cut each other at right angles (that is, their tangents at the point of intersection are perpendicular to each other). Compute, First.—The length of the common chord. Second.—The length of the part of the line of centres intercepted between the centres of the two circles. 7. The sides of a triangle are 3 meters, 5 meters, and 6 meters, respectively. Compute to within a centimeter, First.—The segments cut off on each side by the bisectrix of the opposite angle. Second.—The segments determined by the points of contact of the inscribed circle. Third.—The lengths of the three bisectrices. Fourth. — The segments of each side determined by the correspond ing altitudes. Fifth.—The three altitudes. 8. Given in the straight line AC, AB = 4 inches, BC = 5 inches, and in the line A'C', parallel to AC, A'B' = 1.24 inches, and B'C' = 3.10 inches : discover whether AA', BB', and CC', meet in the same point. 9. Given the side of an equilateral triangle equal to 10 feet : find its area. 11. 10. Given the area of an equilateral triangle equal to 36 square feet : find its side to within .ooi of a foot. Given one of the equal sides of an isosceles triangle equal to 10 feet, and one of the equal angles equal to one-third of a right angle: find the area of the triangle. 12. Given the sum of the squares of the distances of a point, P, from two points, A and B (12 inches apart), equal to 200 square inches : find the radius of the circle which is the locus of P. 13. The length of a tangent, AB, to a circle (whose radius is 200 feet) from the point of contact, A, to the point B is 100 feet. If this tangent is divided into four equal parts, find the lengths of the perpendiculars erected to it at the three points of division, and terminating in the circumference. 14. With the same data as in Problem 13, determine the external portions of the secants from the three points of division which pass through the centre of the circle. BOOK IV. OF REGULAR POLYGONS, AND THE MEASUREMENT OF THE CIRCLE. DEFINITION. A polygon which is at once equiangular and equilateral, is called a regular polygon. Regular polygons may have any number of sides. The equilateral triangle is one of three sides ; and the square, one of four. PROPOSITION I. THEOREM. D F B Two regular polygons of the same number of sides are similar figures. For example, let ABCDEF, abcdef, be two regular hexagons. The sum of all the angles is the same in each figure, being equal to eight right angles (Book I., Prop. XXX.). The angle A is the sixth part of that sum ; so is the angle a; hence, the angles A and a are equal; and for the same reason, the same is true of the angles B and b, C and c, and so on. Again, since from the nature of the polygons, the sides AB, BC, CD, etc., are equal, and likewise the sides ab, bc, cd, etc., it is plain that AB : ab :: BC : bc :: CD : cd; hence, the two figures in question have their angles equal and their homologous sides proportional; therefore they are similar (Book III., Def. 2). Cor. The perimeters of two regular polygons having the same number of sides are to each other as their homologous sides, and * h their surfaces are as the squares of these same sides (Book III., Prop. XXIX.). SCHOLIUM. The angle of a regular polygon is determined by the number of its sides (Book I., Prop. XXX.). PROPOSITION II. THEOREM. B G A circle may be circumscribed about any regular polygon; and a circle may be inscribed in any regular polygon. Let ABCDE, etc., be a regular polygon ; describe a circle through the three points A, B, C; O being the centre, and OP the perpendicular let fall from it on the middle of the side BC; join AO and OD. If the quadrilateral OPCD be placed on the quadrilateral OPBA, they will coincide, for the side OP is. common; the angle H OPC = OPB, being right; hence, the side PC will fall along its equal PB, and the point C will fall on B. Besides, from the nature of the polygon, the angle PCD = PBA, hence, CD will fall along BA, and since CD = BA, the point D will fall on A, and the two quadrilaterals will entirely coincide. The distance OD is therefore equal to AO; and consequently the circle which passes through the three points A, B, C, will pass also through the point D: by similar reasoning it may be shown that the circle which passes through the three vertices B, C, D, will pass through the vertex E, and so of all the rest; hence, the circle which passes through the three points A, B, C, passes through the vertices of all the angles of the polygon, which is therefore inscribed in this circle. Again, in reference to this circle, all the sides AB, BC, CD, etc., are equal chords ; they are therefore equally distant from the centre (Book II., Prop. VIII.); hence, if from the point O, as a centre, with the radius OP, a circle be described, it will touch the side BC and all the other sides of the polygon, each at its middle point, and the circle will be inscribed in the polygon, or the polygon circumscribed about the circle. SCHOLIUM. The point O, the common centre of the inscribed and |