NOTE. The following alternative proof of Proposition 30 removes the necessity of distinguishing between the major and minor arcs cut off by the chords AD, BD. PROPOSITION 30. [ALTERNATIVE PROOF.] The construction being made as before, we may proceed thus : Proof. In the A8 ACD, BCD, Constr. Because and CD is common; I. 4. ... the arc DB = the arc DA: that is, the arc ADB is bisected at D. Q.E.F. III. 26. QUESTIONS FOR REVISION. 1. When is a straight line said (i) to meet, (ii) to cut, (iii) to touch, the circumference of a circle ? 2. When are circles said to touch one another? Distinguish between internal and exiernal contact. 3. What theorems have been so far proved by Euclid regarding (i) circles which cut one another, (ii) circles which touch one another? 4. If two unequal circles are concentric, shew that one must lie wholly within the other. 5. Shew how to divide the circumference of a circle into three, four, or six equal parts. 6. Enunciate the propositions so far proved by Euclid relating to the properties of a tangent to a circle. PROPOSITION 31. THEOREM. The angle in a segment greater than a semicircle is less than a right angle. The angle in a segment less than a semicircle is greater than a right angle. Let ABCD be a circle, of which BC is a diameter, and E the centre; and let AC be a chord dividing the circle into the segments ABC, ADC, of which the segment ABC is greater, and the segment ADC is less than a semicircle. Then (i) the angle in the semicircle BAC shall be a right angle ; (ii) the angle in the segment ABC shall be less than a rt. angle; (iii) the angle in the segment ADC shall be greater than a rt. angle. Construction. In the arc ADC take any point D; Join BA, AD, DC, AE; and produce BA to F. I. Def. 15. the EAB=the _ EBA. I. 5. And because EA= EC, .:the EAC = the 2 ECA. ... the whole - BAC = the sum of the 4S EBA, ECA: but the ext. L FAC = the sum of the two int. 48 CBA, BCA; ::. the BAC = the L FAC; the - BAC, in the semicircle BAC, is a rt. angle. L (ii) In the A ABC, because the sum of the < ABC, BAC is less than two rt. angles ; I. 17. and of these, the < BAC is a rt. angle ; Proved. ... the ABC, which is the angle in the segment ABC, is less than a rt. angle. (iii) Because ABCD is a quadrilateral inscribed in the O ABC, .'. the opp. _* ABC, ADC together=two rt. angles ; III. 22. and of these, the _ ABC is less than a rt. angle: Proved. ... the 2 ADC, which is the angle in the segment ADC, is greater than a rt. angle. Q.E.D. EXERCISES. 1. A circle described on the hypotenuse of a right-angled triangle as diameter, passes through the opposite angular point. 2. A system of right-angled triangles is described upon a given straight line as hypotenuse ; find the locus of the opposite angular points. 3. A straight rod of given length slides between two straight rulers placed at right angles to one another; find the locus of its middle point. 4. Two circles intersect at A and B; and through A two diameters AP, AQ are drawn, one in each circle: shew that the points P, B, Q are collinear. [See Def. p. 110.] 5. A circle is described on one of the equal sides of an isosceles triangle as diameter. Shew that it passes through the middle point of the base. 6. Of two circles which have internal contact, the diameter of the inner is equal to the radius of the outer. Shew that any chord of the outer circle, drawn from the point of contact, is bisected by the circumference of the inner circle. 7. Circles described on any two sides of a triangle as diameters intersect on the third side, or the third side produced. 8. Find the locus of the middle points of chords of a circle drawn through a fixed point. Distinguish between the cases when the given point is within, on, or without the circumference. 9. Describe a square equal to the difference of two given squares. 10. Through one of the points of intersection of two circles draw a chord of one circle which shall be bisected by the other. 11. On a given straight line as base a system of equilateral four-sided figures is described : find the locus of the intersection of their diagonals. NOTES ON PROPOSITION 31. Note 1. The extension of Proposition 20 to straight and reflex angles furnishes a simple alternative proof of the first theorem contained in Proposition 31, namely, The angle in a semicircle is a right angle. E B For, in the adjoining figure, the angle at the centre, standing on the arc BHC, is double the angle BĂC at the Oce, standing on the same arc. H Now the angle at the centre is the straight angle BEC; and a straight angle=two rt. angles ; Q. E. D. NOTE 2. From Proposition 31 we may derive a simple practical solution of Proposition 17, namely, To draw a tangent to a circle from a given external point. Let BCD be the given circle, and A the given external point. İt is required to draw from A a tangent to the O BCD. Find E, the centre of the given circle, and join AE. А! On AE describe the semicircle ABE, to cut the given circle at B. Join AB. Then AB shall be a tangent to the O BCD. For the L ABE, being in a semicircle, is a rt. angle. III. 31. :. AB is drawn at rt. angles to the radius EB, from its extremity B; :: AB is a tangent to the circle. III. 16. Q. E. F. Since the semicircle might be described on either side of AE, it is clear that there will be a second solution of the problem, as shewn by the dotted lines of the figure. QUESTIONS FOR REVISION AND NUMERICAL EXERCISES. 1. Define an arc, a chord, a segment of a circle. When are segments of circles said to be similar to one another? 2. Enunciate propositions which give the properties of chords of a circle in relation to the centre. 3. Prove that in a circle whose diameter is 34 inches, a chord 30 inches in length is at a distance of 8 inches from the centre. 4. In a circle a chord 2 feet in length stands at a distance of 5 inches from the centre : shew that the diameter of the circle is 2 inches longer than the chord. 5. What must be the length of a chord which is 1 foot distant from the centre of a circle, if the diameter is 2 yards 2 inches ? 6. Two parallel chords of a circle, whose diameter is 13 inches, are respectively 5 inches and 1 foot in length : shew that the distance between them is 8} inches, or 31 inches. 7. Two circles, whose radii are respectively 26 inches and 25 inches, intersect at two points which are 4 feet apart. Shew that the distance between their centres is 17 inches. 8. The diameters of two concentric circles are respectively 50 inches and 48 inches : shew that any chord of the outer circle which touches the inner must be 14 inches in length. 9. Of two concentric circles the diameter of the greater is 74 inches, and any chord of it which touches the smaller circle is 70 inches in length: shew that the diameter of the smaller circle is 2 feet. 10. Two circles of diameters 74 and 40 inches respectively have a common chord 2 feet in length: shew that the distance between their centres is 51 inches. 11. The chord of an arc is 24 inches in length, and the height of the arc is 8 inches ; shew that the diameter of the circle is 26 inches. 12. AB is a line 20 inches in length, and C is its middle point. On AB, AC, CB semicircles are described. Shew that if a circle is inscribed in the space enclosed by the three semicircles its radius must be 3} inches. |