THEOREM VIII.

193. Through three points not in the same straight line, a circumference of a circle can be passed.

Let A, B, and C be any three points not in the same straight line.

To prove that through A, B, and C, a circumference of a can be passed.

Draw AB and BC, and at their middle points let the s EM and DP be erected.

AB and BC are not in the same straight line;

thes EM and DP meet in some point, as O.

Now, O is equally distant from A and B; also from B and C;

(53)

... O is equally distant from A, B, and C, and a circumference with OA as a radius passes through these points.

Q. E. D.

THEOREM IX.

194. A straight line perpendicular to a radius at its extremity

is a tangent to the circle.

Let AB be to the radius OP at P.

A

P

B

To prove that AB is a tangent to the ○ at the point P.

From the centre draw any oblique line, as OC.

the point C is without the circumference, and all points in AB, except P, are without the circumference;

AB is a tangent to the O at P. (177) Q. E. D.

195. COR.-A straight line tangent to a circle is perpendicular to the radius drawn to the point of contact.

THEOREM X.

196. Two parallel secants intercept equal arcs.

Let the Is AB and CD intercept the arcs AC and BD.

E

To prove that arc AC = arc BD.

Suppose the radius OE to be drawn to AB and CD.

197. SCHOLIUM.-This proposition is true for any position of the parallels; hence it is true if one or both become tangents; and the straight line which joins the points of contact of two parallel tangents is a diameter.

RELATIVE POSITION OF CIRCLES.

THEOREM XI.

198. If two circles cut each other, the straight line joining their centres bisects their common chord at right angles.

Let AB be a common chord of two Os which cut each other, and OC join the centres O and C.

and

O is equally distant from A and B,

Cis equally distant from A and B;

OC is to AB at its middle point. (55) Q. E. D.

199. COR.-If two circles touch each other, either externally or internally, the point of contact is in the line joining their centres.

THEOREM XII.

200. If two circles cut each other, the distance between their centres is less than the sum and greater than the difference of their radii.

Let O and C be the centres of two Os whose circumferences cut each other at A and B, and draw the radii OA and CA.

To prove that OC <OA + CA, and OC> OA — CA.

The point A does not lie in OC;

(199)

201. COR. 1.-If two circles touch each other externally, the distance between their centres equals the sum of their radii.

202. COR. 2.-If two circles are wholly exterior to each other, the distance between their centres is greater than the sum of their radii.