is outside the circumference O; therefore the straight line A B has only one point in common with the circumference O: that is, A B is tangent to that circumference, W. W. T. B. D. Inversely

A secant is oblique to the radius terminating at a point of section. COROLLARY. From a point without a circumference there may be two tangents thereto.

THEOREM 55.

Conversely, a tangent to a circumference is perpendicular to the radius terminating at the point of tangence.

Let A B be a tangent to the circumference O at the point B.

B C

Let the point of tangence be joined to the centre O, then O B, being a radius, will be shorter than O C joining the centre to any other point of A B, all these points being outside the circumference [Def. 27]. Then O B is the shortest of all straight lines joining the point O to A B; therefore O B is perpendicular to A B [11 i], which W. T. B. D. Inversely

A straight line oblique at the extremity of a radius, is secant to the circumference.

COROLLARY I. Through a point of a circumference, there cannot be more than one tangent to the circumference (9).

COROLLARY II. The tangents at the extremities of a diameter, are parallel to each other, and conversely.

COROLLARY III. The arcs of a circumference intercepted between a tangent and a parallel secant, or tangent, are equal to each other.

THEOREM 56.

The perpendicular raised on a tangent through the point of tangence passes through the centre of the circumference.

Let A B be a tangent at the point B to the circumference O, and let BC be a perpendicular on A B at the same point.

B

If the centre O be joined to the point of tangence, the radius O B will be perpendicular on A B [55], and consequently will coincide with B C [9]; therefore, B C will pass through the centre, W. W. T. B.D. Inversely

If the perpendicular drawn through the point of a straight line in common with a circumference does not pass through the centre, the straight line is secant to the circumference.

COROLLARY I. The chord joining the points of tangence of two tangents drawn to a circumference from a point without it, is less than the diameter.

COROLLARY II. The arc convex to a point from which two tangents, or two equal secants, are drawn to a circumference, is less than the arc concave to the same points.

Scholium. The chord perpendicular to a tangent through the point of tangence is a diameter.

THEOREM 57.

Two tangents to a circumference drawn from a point without it are equal to each other.

Let A B and A C be two tangents from the point A to the circum. ference O.

B

N

M

Let OB and OC be two radii terminating at the points of tangence, and A N M a secant passing through the centre O. Let

the plane containing the figure be folded upon A M. Because M N is a diameter [Def. 26], the arc M BN will coincide with the arc MCN [39]. Also the point B will coincide with the point C; for otherwise, the radius O B would take a position O D, or O E, on either side of O C; the tangent A B would become A D, or A E, and cut the radius O C in a point I, and O C, being perpendicular on A C [55], would be an oblique on A D [10], whilst O D, or O E, would be perpendicular on A D [55]. The oblique O I on A D, which is less than a radius, would then be less than the radii perpendicular O D, or O E, from the same point, which is impossible [11]; and the further D or E should be removed from C, the less O I would become: therefore D, or E, coincide with C and A B with AC; that is, A B is equal to A C, which W. T. B. D. Inversely

If one of two unequal straight lines drawn to a circumference from a point without it be a tangent, the other is a secant.

COROLLARY I. From a point without a circumference there cannot be more than one tangent to it on the same side of the centre secant from the same point.

COROLLARY II. The tangent is less than any secant drawn from the same point without the circumference, and greater than its outer part. COROLLARY III. From a point without a circumference there cannot be more than two equal straight lines drawn thereto.

COROLLARY IV. The tangents drawn to a circumference from a point without it are symmetric to the centre secant from the same point. COROLLARY V. The straight line drawn through the centre from a point without the circumference, bisects the convex and concave arcs to that point.

H

SECTION III.

ON TANGENT CIRCUMFERENCES.

DEFINITIONS.

30. Two circumferences which have no point in common are internal to one another, or simply internal, when no part of the one is outside the other; they are external to one another, or simply external, when no part of the one is within the other.

31. Two circumferences are concentric to each other, or simply concentric, when they have the same centre, and excentric when they have distinct centres.

32. Two circumferences which have one point in common, and no more, are tangent to one another, or simply tangent; their common point is the point of tangence, or of con

tact.

Two circumferences may be internally, or externally, tangent to one another.

33. Two distinct circumferences which have more than one point in common are secant to one another, or simply secant; the points in common are their points of section, and the straight line joining them is their common chord.

Two secant circumferences cannot have more than two points of section since they would then coincide.

It is clear that two circumferences can have but five different positions relatively to one another: they may be internal, internally tangent, secant, externally tangent, or external to one another.

34. The straight line which joins the centres of two circumferences, is their centre line.