Proposition 18. Theorem. 248. An angle formed by a tangent and a secant is measured by one-half the difference of the intercepted arcs. E H Hyp. Let AC, AB be a tangent and a secant intersecting at A. To prove A is measured by (arc BHE- arc DE). 249. COR. The angle formed by two tangents is measured by one-half the difference of the intercepted arcs. EXERCISES. 1. Two tangents AB, AC are drawn to a circle; D is any point on the circumference outside the triangle ABC: show that the sum of the angles ABD and ACD is constant. 2. If a variable tangent meets two parallel tangents it subtends a right angle at the centre. QUADRILATERALS. Proposition 19. Theorem. 250. If the opposite angles of a quadrilateral are supplementary, the quadrilateral may be inscribed in a circle. Hyp. Let ABCD be a quadrilateral in which ZB+ 2D=2 rt. s. To prove the pts. A, B, C, D are in the same O. Proof. Through the three pts. A, B, C describe a O. B A E will cut AD, or AD produced, at some other pt. than D. Let E be this pt. Join EC. Because the quadrilateral ABCE is inscribed in a ○, .. ZABC+AEC 2 rt. s. The opp. 48 of an inscribed quad. are supplementary (242). that is, an ext. ▲ of a ▲ = an int. opp. /, which is im possible. (98) .. the circle which passes through A, B, C, must pass through D. 251. DEF. Q. E.D. Points which lie on the circumference of a circle are called concyclic. A cyclic quadrilateral is one which is inscribed in a circle. EXERCISE. If two opposite sides of a cyclic quadrilateral be produced to meet, and a perpendicular be let fall on the bisector of the angle between them from the point of intersection of the diagonals: prove that this perpendicular will bisect the angle between the diagonals, Proposition 20. Theorem. 252. In any quadrilateral circumscribing a circle, the sum of one pair of opposite sides is equal to the sum of the other pair. Hyp. Let ABCD be a quadrilateral circumscribing a O. To prove AB + CD = AD + BC. Proof. From the centre O draw the radii to the pts. of contact E, F, G, H, and draw OB. Then rt. A OBE=rt. ▲ OBF, (110) E Similarly, EA = HA, GD = HD, GC = FC. Adding these four equations, we have or EBEA+ GD + GC = FB + HA + HD + FC, AB+ CD = BC + AD. G Q.E.D. EXERCISES. 1. The line joining the middle points of two parallel chords of a circle passes through the centre. 2. The chords that join the extremities of two equal arcs of a circle towards the same parts are parallel. 3. The sum of the angles subtended at the centre of a circle by two opposite sides of a circumscribed quadrilateral is equal to two right angles, 4. An isosceles triangle has its vertical angle equal to the exterior angle of an equilateral triangle. Prove that the radius of the circumscribing circle is equal to one of the equal sides of the given triangle. 5. Prove that the radius of the circle inscribed in an equilateral triangle is equal to one-third of the altitude of the triangle. 6. The quadrilateral ABCD is inscribed in a circle; AB, DC produced meet in E: prove that the triangles ACE, BDE, and also the triangles ADE, BCE, are mutually equiangular. 7. The quadrilateral ABCD is inscribed in a circle; AB, DC produced meet in E, and BC, AD produced meet in F; the sides AB, BC, CD subtend arcs of 120°, 70°, 80°, respectively: find the number of degrees in the angles AED and AFB. 8. In the circumscribed quadrilateral ABCD, the angles A, B, C are 110°, 95°, 80°, respectively, and the sides AB, BC, CD, DA touch the circumference at the points E, F, G, H respectively: find the number of degrees in each angle of the quadrilateral EFGH. 9. If two opposite sides of an inscribed quadrilateral are equal, prove that the other two sides are parallel. PROBLEMS OF CONSTRUCTION. 253. Hitherto, our investigations have been purely theoretical, and have been confined to the demonstration of certain properties of figures, assumed to exist, satisfying certain conditions; and our figures have been assumed to be constructed under these conditions, although no methods of constructing them have been given. Indeed, the precise construction of the figures was not necessary, as they were required only as aids in following the demonstration of principles. The constructions were only hypothetical. We now apply these principles to determine methods by means of which such figures are to be approximately drawn; for, owing to the imperfection of our instruments, the ideal state contemplated in theoretical Geometry cannot be attained by them. The determination of the method of constructing a given figure with given instruments is called a problem; and the solution of a problem requires us to show how the construction can be affected by the use of the given instruments, and to prove that the construction is correct. The solution of a problem depends on the instruments that are used. The more restricted the choice of instruments, the more limited will be the problems which can be solved by their use; and the more difficult will be the solution of many that are thus solved. It is the recognized convention of Elementary Geometry that the only instruments to be employed are the ruler and compasses, with the use of which the student should become familiar. The ruler is used for drawing and producing straight lines, and the compasses for describing circles and for the transference of distances; the straight line and the circumference being the only lines treated of in this subject. This convention is embodied in the three postulates given in (45). Problems, though important as applications of geometric truths, form no part of the chain of connected truths embodied in the theorems of Geometry, so that, though they may be studied advantageously in connection with the theorems on which they directly depend, they are not a necessary part of the pure science of Geometry.* * Elements of Plane Geometry, Association for the improvement of Geometric Teaching, p. 69. |