THEOREM XIX. 383. The angle in a segment is greater than, equal to, or less than, a right angle, according as the arc of the segment is less than, equal to, or greater than, a semicircle. HYPOTHESIS. Let AD be a diameter of a circle whose center is 0. Take B and C points on the same circle. CONCLUSIONS. I. ¥ ACD = rt. X, being half the straight angle AOD. (375. An angle at the center is double the inscribed angle on the same arc.) (143. Any two angles of a triangle are together less than a straight angle.) since III... ABC > rt. 4, × ADC + × ABC = st. ¥. (379. Opposite angles of an inscribed quadrilateral are supplemental.) 384. By 33, Rule of Inversion, A segment is less than, equal to, or greater than, a semicircle, according as the angle in it is greater than, equal to, or less than, a right angle. THEOREM XX. 385. If two chords intersect within a circle, an angle formed and its vertical are each equal to half the angle at the center standing on the sum of the arcs they intercept. HYPOTHESIS. Let the chords AC,BD intersect at F within the circle. CONCLUSION. BFC = half ¥ at center standing on (arc BC + arc DA). PROOF. Join CD. X BFC = X BDC + X DCA. × × (173. The exterior angle of a triangle is equal to the sum of the opposite interior angles.) But 2 ¥ BDC = at center on BC, and 2 × DCA = at center on DA; (375. An angle at the center is double the inscribed angle upon the same arc.) .. 2 × BFC = × at center on arc (BC + DA). (367. A sum of two arcs subtends an angle at the center equal to the sum of the angles subtended by the arcs.) EXERCISES. 71. The end points of two equal chords of a circle are the vertices of a symmetrical trapezoid. 72. Every trapezoid inscribed in a circle is symmetrical. THEOREM XXI. 386. An angle formed by two secants is half the angle at the center standing on the difference of the arcs they intercept. HYPOTHESIS. Let two lines from F cut the circle whose center is O, in the points A, B, C, and D. Doubling both sides, at O on arc CD = 2 4 F + ¥ at ◇ on arc AB ; .'. twice F = difference of s at O on arcs CD and AB. IV. Tangents. 387. A line which will meet the circle in one point only is said to be a Tangent to the circle. 388. The point at which a tangent touches the circle is called the Point of Contact. THEOREM XXII. 389. Of lines passing through the end of any radius, the perpendicular is a tangent to the circle, and every other line is a HYPOTHESIS. A radius OP perpendicular to PB, oblique to PC. CONCLUSIONS. I. PB is a tangent at P. II. PC is a secant. PROOF. (I.) The sect from 0 to any point on PB, except P, is > OP, (150. The perpendicular is the least sect between a point and a line.) ... every point of PB except P is outside the circle. (321. A point is without a circle if its sect from the center is greater than the radius.) OP, PROOF. (II.) A sect perpendicular to PC is less than the oblique 390. COROLLARY I. One and only one tangent can be drawn to a circle at a given point on the circle. 391. COROLLARY II. To draw a tangent to a circle at a point on the circle, draw the perpendicular to the radius at the point. 392. COROLLARY III. The radius to the point of contact of any tangent is perpendicular to the tangent. 393. COROLLARY IV. The perpendicular to a tangent from the point of tangency passes through the center of the circle. 394. COROLLARY V. The perpendicular drawn from the center to the tangent passes through the point of contact. ON THE THREE RELATIVE POSITIONS OF A LINE AND A CIRCLE. 395. COROLLARY VI. A line will be a secant, a tangent, or not meet the circle, according as its perpendicular from the center is less than, equal to, or greater than, the radius. 396. By 33, Rule of Inversion, The perpendicular on a line from the center will be less than, equal to, or greater than, the radius, according as it is a secant, tangent, or non-meeter. THEOREM XXIII. 397. An angle formed by a tangent and a chord from the point of contact is half the angle at the center standing on the intercepted arc. F HYPOTHESIS. AB is tangent at C, and CD is a chord of © with center at O. |