extended chord GH is, therefore, equal to all the three chords BE, CF, and DG. It is obvious, that the same train of reasoning may be pursued to any number of equal arcs. PROP. XXV. THEOR. If, from the end of an arc, a straight line equal to the radius of the circle, be inflected to a diameter extending through the other end, and be produced to meet the circumference; it will intercept, from the remoter extremity of the diameter, an arc which is triple of the first arc. Let AB be an arc of a circle, and D a point in the extended diameter FB, such that DA is equal to the radius CB; on producing DA to meet the circumference in E, the arc FE, thus intercepted, is triple of AB. G E If the point E lie between F and A; join AC, and draw AG parallel to DF. Because AD is equal to the radius AC, the angle ACB is equal to ADC (I. 8.); and DC being parallel to AG, the angle ADC is equal to EAG (I. 25.), and consequently EAG is equal to ACB. But an angle at the centre on the same F C B base GE would be double of EAG or ACB (III. 19.); wherefore the arc GE is double of AB (III. 20. cor.), and GF being equal to AB (III. 22.), the whole arc FE is triple of AB. Again, if the point E lie beyond FA. Draw EG paral lel to BF. And AD being equal to AC, the angle ACB is equal to ADC; but ADC is equal to the interior angle AEG, consequently the central angle ACB is equal to AEG at the circumference; wherefore the arc GFA is double of AB, and GFAB its triple; add to the one F A C DB E side, and take away from the other, the equal arcs BE and FG, and there results the arc FAE triple of AB. PROP. XXVI. THEOR. The angle in a semicircle is a right angle, the angle in a greater segment is acute, and the angle in a smaller segment is obtuse. Let ABD be an angle in a semicircle, or that stands on the semicircumference AED; it is a right angle. For ABD, being an angle at the circumference, is half of the angle at the centre on the same base AED (III. 19.); it is, therefore, half of the angle ACD formed by the opposite portions CA, CD of the diameter, or half of two right angles, and is consequently equal to one right angle. B D Again, let ABD be an angle in a segment greater than a semicircle, or which stands on a less arc AED than the semicircumference; it is an acute angle. on the arc ABD, which is greater than a semicircumference, and is the base of an angle at the centre, the reverse of ACD, and greater, therefore, than two right angles; AED is hence an obtuse angle. Cor. From the remarkable property, that the angle in a semicircle is a right angle, may be derived an elegant method of drawing perpendiculars. PROP. XXVII. THEOR. If a circle be described on the radius of another circle, any straight line drawn from the point where they meet to the outer circumference is bisected by the interior one. Let AEC be a circle described on the radius AC of the circle ADB, and AD a straight line drawn from A to terminate in the exterior circumference; the part AE in the smaller circle is equal to the part ED intercepted between the two circumferences. B For join CE, CD. And because AEC is a semicircle, the angle contained in it is a right angle (III. 25.); con sequently the straight line CE, drawn from the centre C, is perpendicular to the chord AD, and therefore bisects it (III. 5.) PROP. XXVIII. THEOR. The perpendicular at the extremity of a diameter is a tangent to the circle, and is the only tangent which can be applied at that point. Let ACB be the diameter of a circle, to which the straight line EBD is drawn at right angles from the extremity B; it will touch the circumference at that point. For CB, being perpendicu lar, is the shortest distance of the centre C from the straight line EBD (I. 22.); wherefore every other point in this line is farther from the centre than B, and consequently falls without the circle. But the perpendicular EBD is the only straight line which can be drawn through the point B that will not cut the circle. For if HBF were such a line, the perpendicular CG, let fall upon it from the centre, would be less than CB (I. 22.) and would therefore lie within the circle; consequently HBG, being extended, would again meet the circumference before it effected its escape. Cor. Hence a straight line drawn from the point of contact at right angles to a tangent, must be a diameter, or pass through the centre of the circle. Schol. The nature of a tangent to the circle is easily discovered from the consideration of limits. For suppose the H straight line DE, extending both ways, to turn about the extremity B of the diameter AB; it will cut the circle first on the one side of AB, and afterwards on the other. But the arc AH being less than a semicircumference, the angle HBA which the line D'E' makes with the diameter is acute (III. 25.); and for the same reason, the angle KBA is acute, and consequently its adjacent angle D'BA is obtuse. Thus the revolving line DE, when K it meets the semicircumference AHB, makes an acute angle with the diameter; but when it comes to meet the opposite semicircumference, it makes an obtuse angle. In passing, therefore, through all the intermediate gradations from minority to majority, the line DE must find a certain individual position in which it is at right angles to the diameter, and cuts the circle neither on the one side nor the other. A similar inference might be derived from Prop. 22. of this Book; one of the parallel chords being supposed to contract, until its extreme points are about to coalesce in the position of the tangent. PROP. XXIX. THEOR. If from the point of contact a straight line be drawn to cut the circumference, the angles which it makes with the tangent are equal to those in the alternate segments of the circle. Let CD be a tangent, and BE a straight line drawn |