PROPOSITION XXIX. THEOREM. In equal circles, the chords, which subtend equal arcs, must be equal. Let ABC, DEF be equal circles, and let BC, EF be chords subtending the equal arcs BGC, EHF. Then must chord BC chord EF. Take the centres K, L. Then, if ABC be applied to o DEF, arc BGC=arc EHF, .. C will coincide with F. .. chord BC must coincide with and be equal to chord EF. Q. E. D. Ex. 1. The two straight lines in a circle, which join the extremities of two parallel chords, are equal to one another. Ex. 2. If three equal chords of a circle, cut one another in the same point, within the circle, that point is the centre. NOTE 4. On the Symmetrical properties of the Circle with regard to its diameter. The brief remarks on Symmetry in pp. 107, 108 may now be extended in the following way: A figure is said to be symmetrical with regard to a line, when every perpendicular to the line meets the figure at points which are equidistant from the line. Hence a Circle is Symmetrical with regard to its Diameter, because the diameter bisects every chord, to which it is perpendicular. B Further, suppose AB to be a diameter of the circle ACBD, of which O is the centre, and CD to be a chord perpendicular to AB. Then, if lines be drawn as in the diagram, we know that AB bisects (1.) The chord CD, III. 1. (2.) The arcs CAD and 'CBD, III. 26. (3.) The angles CAD, COD, CBD, and the reflex These Symmetrical relations should be carefully observed, because they are often suggestive of methods for the solution of problems. :: AD=CD, and DB is common, and ▲ ADB = 2 CDB, But, in the same circle, the arcs, which are subtended by equal chords, are equal, the greater to the greater and the less to the less; and BD, if produced, is a diameter, III. 28. .. each of the arcs BA, BC, is less than a semicircle, and .. arc BA=arc BC. Thus the arc ABC is bisected in B. Q. E. F. Ex. If, from any point in the diameter of a semicircle, there be drawn two straight lines to the circumference, one to the bisection of the circumference, and the other at right angles to the diameter, the squares on these two lines are together double of the square on the radius In a circle, the angle in a semicircle is a right angle; and the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle. B Let ABC be a O, O its centre, and BC a diameter. Then must the ▲ in the semicircle BAC be a rt. 4, and ▲ in segment ABC, greater than a semicircle, less than a rt. 4, and in segment ADC, less than a semicircle, greater than a rt. 4. First, BO=AO, .. 4 BAO= 2 ABO; I. A. I. 32. I. A. I. 32. .. sum of 48 COA, BOA=twice sum of 4 8 BAO, CAO, that is, two right angles twice / BAC. .. L BAC is a right angle. Next, BAC is a rt. 4, .. 4 ABC is less than a rt. 4. Lastly, sum of 4 s ABC, ADC=two rt. 4 s, and ABC is less than a rt. L .. ADC is greater than a rt. 4. NOTE. For a simpler proof see page 178. I. 17. III. 22. Q. E. D. Ex. 1. If a circle be described on the radius of another circle as diameter, any straight line, drawn from the point, where they meet, to the outer circumference, is bisected by the interior one. Ex. 2. If a straight line be drawn to touch a circle, and be parallel to a chord, the point of contact will be the middle point of the arc cut off by the chord. Ex. 3. If, from any point without a circle, lines be drawn touching it, the angles contained by the tangents is double of the angle contained by the line joining the points of contact, and the diameter drawn through one of them. Ex. 4. The vertical angle of any oblique-angled triangle inscribed in a circle is greater or less than a right angle, by the angle contained by the base and the diameter drawn from the extremity of the base. Ex. 5. If, from the extremities of any diameter of a given circle, perpendiculars be drawn to any chord of the circle that is not parallel to the diameter, the less perpendicular shall be equal to that segment of the greater, which is contained between the circumference and the chord. Ex. 6. If two circles cut one another, and from either point of intersection diameters be drawn, the extremities of these diameters and the other point of intersection lie in the same straight line. Ex. 7. Draw a straight line cutting two concentric circles, so that the part of it which is intercepted by the circumference of the greater may be twice the part intercepted by the circumference of the less. Ex. 8. Describe a square equal to the difference of two given squares. Ex. 9. If from the point in which a number of circles touch each other, a straight line be drawn cutting all the circles, shew that the lines, which join the points of intersection in each circle with its centre, will all be parallel. |