PROP. XVIII. THEOR.

Parallel chords intercept equal arcs of a circle.

Let the chord AB be parallel to CD; the intercepted arc AC is equal to BD.

For join AD. And because the straight lines AB and CD are parallel, the alternate angles BAD and ADC are equal (I. 22.); wherefore these angles, having their vertices in the circumference of the circle, must stand on equal arcs (III. 16. cor.),

B

D

and consequently the arcs AC and BD are equal to each other.

Cor. Hence, conversely, the straight lines which intercept equal arcs of a circle are parallel; and hence another mode of drawing a parallel through a given point to a given straight line.

PROP. XIX. 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. 15.); it is, therefore, half of the angle ACD formed by the diverging of the opposite portions CA, CD of the diameter, or

B

D

or half of two right angles, and is consequently equal to one right angle.

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.

For join CA, CD. The angle ABD is half of the central angle ACD, which is evidently less than two right angles; wherefore ABD is less than

one right angle, or it is acute.

But the angle AED, in the smaller segment, is obtuse. For AED stands on the arc ABD, which is greater than a semicircumference, and is the base of an angle at the

E

centre, the reverse of ACD, and greater, therefore, than two right angles; AED is hence an obtuse angle.

Cor. Hence conversely the arc which contains a right angle must be a semicircle.

Schol. From the remarkable property, that the anglé in a semicircle is a right angle, may be derived an elegant method of drawing perpendiculars.

PROP. XX. THEOR.

The perpendicular at the extremity of a diameter is a tangent to the circle, and 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 extre

mity 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. 17.); wherefore every other point in this line is farther from the centre than B,

and consequently falls without the circle.

B

I

But EBD, drawn at right angles to the diameter, is the only straight line which can pass through the point B and not cut the circle.

For were HBF such a line, the perpendicular CG let fall upon it from the centre, would be less than CB (I. 17.), and must therefore lie within the circle; consequently HBG, being extended, would again meet the circumference.

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.

H

Scholium. The nature of a tangent to the circle is easily discovered from the consideration of limits. For suppose the 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

A

K

E

(III. 19.); 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 it meets the semicir

cumference 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.

PROP. XXI. 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 from the point of contact, cutting the circle into two segments BAE and BFE; the angle EBD is equal to EAB, and the angle EBC to EFB.

For draw BA perpendicular to CD (I. 5. cor.), join AE, and from any point F in the opposite arc, draw FB and FE.

Because BA is perpendicular to the tangent at B, it is a diameter (III. 20. cor.), and consequently AEFB is a semicircle; wherefore AEB is a right angle (III. 19.), and the remaining acute an

gles BAE, ABE of the triangle, being together equal to another right angle, are equal to ABE and EBD,

which compose the right angle ABD.

F

B

D

Take the angle ABE away from both, and the angle BAE

remains equal to EBD.

Again, the opposite angles BAE and BFE of the quadrilateral figure BAEF, being equal to two right angles (III. 17.), are equal to the angle EBD with its adjacent angle EBC; and taking away the equals BAE and EBD, there remains the angle BFE equal to EBC.

Cor. If a straight line meet the circumference of a circle, and make an angle with an inflected line equal to that in the alternate segment, it touches the circle.

Schol. A tangent may be considered as only a secant arrived at its ultimate position, when the two points through which it is drawn come to coincide. Suppose the straight line joining B and F were extended, it would make with the chord BE an angle EBF, equal to what the arc EF subtends from any point in the opposite circumference. But, when the point Fis brought into the situation B, and BF merges into a tangent, the angle EBF passes into EBD, and the angle of the opposite or alternate segment becomes BAE.

PROP. XXII. PROB.

To draw a tangent to a circle, from a given point without it.

Let A be a given point, from which it is required to draw a straight line that shall touch the circle DGH.

Find the centre C(III. 5. cor.),

join AC, and on this as a diameter describe the circle AGCK, cutting the given circle in the points G, K: Join AG, AK; either of these lines is the tangent required.

A

D

H

For join CG, CK. And the angles CGA, CKA, be