When B, moving in the same direction, passes B', the elements change name, those which were formerly the minor becoming the major and vice versa. 106°. Theorem.-An angle at the circle is one-half the corresponding angle at the centre, major corresponding to major and minor to minor. The theorem is thus proved for the minor angles. But since the limiting conditions require only an angle at the circle and an angle at the centre, the theorem remains true while B moves along the circle. And when B passes B' the angle APB becomes the major angle at the circle, and the angle AOB minor becomes the major angle at the centre. the theorem is true for the major angles. .'. Cor. 1. The angle in a given arc is constant. (105°, 4) Cor. 2. Since LAPB LAOB minor, and LAQB = {LAOB major, and. LAOB minor + LAOB major=4 right angles (37°) But LAPB is supplementary to LAQB, LAPB=4BQD. Hence, if one side of a concyclic quadrangle be produced, the external angle is equal to A the opposite internal angle. Cor. 4. Let B come to B'. Then P B (Fig. of 106°) D LAOB' is a straight angle, LAPB' is a right angle. But the arc APB' is a semicircle, (102°, Def.) Therefore the angle in a semicircle is a right angle. 107°. Theorem.-A quadrangle which has its opposite angles supplementary has its vertices concyclic. (Converse of 106°, Cor. 2) ABCD is a quadrangle whereof the LADC is supplementary to LABC; then a circle Proof. If possible let the through A, A LAPC is supplementary to LABC, LAPC=LADC, D C B (106°, Cor. 2) (hyp.) (60°) g.e.d. which is not true. .. the cannot cut AD in any point other than D, Hence A, B, C, and D are concyclic. Cor. 1. The hypothenuse of a right-angled triangle is the diameter of its circumcircle. (88°, 3, Def.; 97°, Def.) Cor. 2. When P moves along the ○ the ▲APC (last figure) has its base AC constant and its vertical angle APC constant. Therefore the locus of the vertex of a triangle which has a constant base and a constant vertical angle is an arc of a circle passing through the end-points of the base. This property is employed in the trammel which is used to describe an arc of a given circle E It consists of two rules (16°) L and M joined at a determined angle. When it is made L 108°. Theorem.~The angle between two intersecting secants is the sum of those angles in the circle which stand 1. If a six-sided rectilinear figure has its vertices concyclic, the three alternate internal angles are together equal to a circumangle. 2. In Fig. 105°, when B comes to Q, BQ vanishes; what is the direction of BQ just as it vanishes? 3. Two chords at right angles determine four arcs of which a pair of opposite ones are together equal to a semicircle. 4. A, B, C, D are the vertices of a square, and A, E, F of an equilateral triangle inscribed in the same circle. What is the angle between the lines BE and DF? between BF and ED? SPECIAL SECANTS--TANGENT. 109°. Let P be a fixed point on the OS and Q a variable' special positions. The first of these is when Q is farthest distant from P, as at Q', and the secant L becomes a centreline. The second is when Q comes into coincidence with P, and the secant takes the position TT' and becomes a tangent. Def. 1.—A tangent to a circle is a secant in its limiting position when its points of intersection with the circle become coincident. That the tangent cannot cut or cross the O is evident. For if it cuts the O at P it must cut it again at some other point. And since P represents two points we would have the absurdity of a line cutting a circle in three points. (94°) Def. 2.—The point where P and Q meet is called the point of contact. Being formed by the union of two points it represents both, and is therefore a double point. From Defs. I and 2 we conclude 1. A point of contact is a double point. 2. As a line can cut a O only twice it can touch a O only once. 3. A line which touches a O cannot cut it. 4. A is determined by two points if one of them is a given point of contact on a given line; or, only one circle can pass through a given point and touch a given line at a given point. (Compare 97°.) 110°. Theorem.-A centre-line and a tangent to the same point on a circle are perpendicular to one another. L' is a centre-line and T a tangent, both to the point P. Then L' is to T. Proof.—. T has only the one point P in common with the O, every point of T except P lies without the O... if O is the centre on the line L', OP is the shortest segment from O to T. OP, or L', is to T. (63°, 1) q.c.d. 'ז Τ Cor. 1. Tangents at the end-points of a diameter are parallel. Cor. 2. The perpendicular to a tangent at the point of contact is a centre-line. (Converse of the theorem.) Cor. 3. The perpendicular to a diameter at its end-point is a tangent. D 111°. Theorem.-The angles between a tangent and a chord from the point of contact are respectively equal to the angles in the opposite arcs into which the chord divides the circle. A TP is a tangent and PQ a chord to the same point P, and A is any point on T' the O. Then and LDPQ is comp. of LQDP, Similarly, the QPT'=LQBP. (77°, 6) q.e.d. |