9. Prob. To draw a tangent to a circle at a given point in the circumference.

SOLUTION.-Let P be the point at which a tangent is to be drawn. Draw the radius OP to the given point of tangency, and produce it any convenient distance beyond the circle. Erect a perpendicular to this line at P, as MT; then is MT a tangent to the circle (172).

M

FIG. 141*.

10. Prob.-To find the centre of a circle whose circumference is known, or of any arc of it.

SUG. The process is given in PART I. Do the work as there directed, and then show upon what proposition in this section it is founded.

SYNOPSIS.

PROP. I. How divide circles and circumferences.

Cor. 1. Bisects angle.

PROP. II. Radius perp.
to chord.

CHORDS.

PROP. III.

PROP. IV.

PROP. V.

PROP. VI.

SECANTS.

TANGENTS.

PARALLELS.

{

Cor. 4. Dist. from centre.

Distance of equal chords from centre.
Equal arcs, and converse.

PROP. VII. Intersect in only two points.

PROP. VIII. If a line intersect in one

point, it intersects
also in another.

PROP. IX. Line perpendicular to

radius at extremity.

Cor. Line oblique to radius at extr.

Cor. Converse.

PROP. X. Parallel secants intercept equal arcs.

PROP. XI. Secant par. to tangent. Cor. Two parallel

Prob. To bisect an arc.

tangents.

Prob. To bisect an angle.

EXERCISES.

Prob. To draw a tangent at a point in circumference.
Prob. To find centre of circumference or arc.

SECTION V.

OF THE RELATIVE POSITIONS OF CIRCUMFERENCES.

PROPOSITION I.

177. Theorem.-All the circumferences which may be passed through three points not in the same straight line coincide, and are one and the same.

DEM.-Let A, B, and C be three points not in the same straight line; then all the circumferences which can be passed through them will coincide. For join the points, two and two, by straight lines, as AB and BC. Bisect these lines with perpendiculars, as DF and EH. Since AB and BC are not in the same straight line, DF and EH will meet when sufficiently. produced, at one and only one point, as O, because they are straight lines Now, every point in FD is equally distant from A and B, and every point in HE is equally distant from B and C (129). Hence O is equally distant from the three points A, B, and C; and, if a circumference be drawn with O as a centre, and a radius AO, it will pass through the three points. Moreover, every circumference passing through these points must have O for its centre, since the centre must be in FD (otherwise it would be unequally distant from A and B), and also in HE (129). But these lines intersect only in O. Also, every circumference with O as its centre, and passing through A, must have AO for its radius. Hence, as all circles having the same centre and the same radius coincide, all those passing through three points, A, B, and C, coincide. Q. E. D.

FIG. 142.

178. COR. 1.-Through any three points not in the same straight line a circumference can be passed.

179. COR. 2.-Three points not in the same straight line determine A circumference as to position and extent; i.e., in all respects.

180. Cor. 3.-Two circumferences can intersect in only two points.

For, if they have three points common, they coincide, and form one and the same circumference.

182. Theorem.-If a straight line be drawn through the centres of two circles, of the intersections of either circumference with that line, the one on the side toward the centre of the other circle is the nearest point in this circumference to that centre, and the one on the opposite side is the farthest point from that centre.

DEM.-Let M and N, or M' and N', be two circumferences whose centres are O and O'. Draw an indefinite line through these centres. Let A and H be the intersections of M or M' with this line, of which A is on the side of M or M' toward the centre O', and

H is on the opposite side. Then is A the nearest point in M or M' to O', and H the

farthest point from O'.

First, To show that A is nearer O' than any other point in the circumference. A will lie between O and O',

in O', or beyond Oʻ. When

A lies between O and O', as in M, let P be any other point in M, and draw OP and O'P. Now OO' being a straight line, is less than OPO', a broken line. Subtracting OA from the former, and its equal OP from the latter, we have AO' < PO'. When A falls at O' the truth is self-evident. When A lies beyond O', as in M', let P be any other point in M', and draw OP and O'P. Now O'P + 00' > OP (= OA). Subtracting OO' from both, we have O'P > OA – 00' (=O'A). Hence, in any case, A is the nearest point in M or M' to O'.

Second, To show that H is the farthest point in M or M' from O'. In either

figure, let P' be any other point in the circumference than H, and draw OP and O'P'. Now, P'O+ 00' > P'O'. But P'O HO. Hence HO+ OO' HO') > P'O'

PROPOSITION IV.

183. Theorem.-When the distance between the centres of two circles is greater than the sum of their radii, the circumferences are wholly exterior the one to the other.

DEM.-Let M and N be the circumferences of two circles whose centres are O and O'. Let OO' be greater than the sum of the radii. Then are M and N wholly exterior the one to the other.

M

A B

FIG. 145.

N

For A, the intersection of M with OO', is between O and O', since OA OO'. Now, by hypothesis, OO' > O'A + BO'. Subtracting OA from both, we have AO' > BO'. Hence, as the nearest point in M is farther from O' than the

circumference of the latter circle, M lies wholly exterior to N. Q. E. D.

184. COR.-Conversely, When two circumferences are exterior the one to the other, the distance between their centres is greater than the sum of their radii.

DEM.-For, join the centres OO' with a straight line. Now the point A where this line cuts the circumference M is the nearest point in this circumference to the centre O'. But, by hypothesis, this (and every other point in cir cumference O) is without circle O'. Hence, AO' > BO'. To each add OA, and OA+ AO' (or OO') > OA + BO'.

PROPOSITION V.

185. Theorem.-When the distance between the centres of two circles is equal to the sum of their radii, the circumferences are tangent to each other externally.

DEM.-Let M and N be two,circumferences, and OO', the distance between their centres, be equal to OC+ O'C', the sum of their radii; then are the circumferences tangent to each other externally.