Proposition 9. Theorem.

214. Two parallel lines intercept equal arcs on the cir cumference.

A st. line to one of two || s is to the other (71).

E

.. arc AE = arc BE, and arc CE = arc DE, (201) ... arc AC arc BD.

(Ax. 3)

CASE II. When AB is a tangent and CD is a secant.

To prove arc CE = arc DE.

Proof. Draw the radius OE to the pt.

of contact E.

Then

OE is to AB (210),

DO

arc EMH arc ENH.

Proof. Draw the secant MN || to AB. M
Then arc ME = arc NE,
arc MH = arc NH.

and

Adding,

} (Case II)

arc EMH arc ENH. Q. E. D.

215. COR. 1. Conversely, if the arcs intercepted by two secants are equal, the secants are parallel.

216. COR. 2. The straight line joining the points of contact of two parallel tangents is a diameter.

Proposition 10. Theorem.

217. Through three given points not in the same straight line, one circumference, and only one, can be drawn.

Hyp. Let A, B, C be the three given pts. not in a st. line.

To prove that one Oce, and only one, can be drawn through A, B, C.

Proof. Join AB, BC.

Bisect AB, BC by the Ls DF, EG.

Since AB, BC are not in the same st. line,

.. the Is DF, EG must meet at some pt. O.

Then, because O is equidistant from A, B, C,

(Нур.)

(76)

(66)

(66)

.. the Oce described with centre O and radius OA will

pass through A, B, C.

Again, only one Oce can be so described.

For if any Oce pass through A, B, C, its centre will be at once in the bisectors DF, EG, and .. at their pt. of intersection. But two st. lines cannot intersect in more than one pt.

..there is only one Oce that can pass through A, B, and C.

Q.E.D.

218. COR. 1. Two circumferences cannot intersect in more than two points.

219. COR. 2. Two circumferences which have three points common coincide.

RELATIVE POSITION OF TWO CIRCLES.

Proposition 11. Theorem.

220. If two circumferences intersect each other, the right line joining their centres bisects their common chord at right angles.

Hyp. Let O, O' be the centres of two Oces which intersect each other; and A, B their pts. of intersection.

To prove that the line 00' bisects AB at rt. s.

A

B

Proof. Because O and O' are each equally distant from

A and B,

.. the line 00' bisects AB at rt s.

(179)

(67)

221. COR. 1. Conversely, the perpendicular bisector of a common chord passes through the centres of both circles.

222. COR. 2. If we suppose the circles to be moved so that the point A approaches the line 00', the pt. B will also approach the line; and since the line 00' is always perpendicular to the mid

G

A

dle of AB (220), the two points

A and B will ultimately come together on the line 00', and be united in a single point common to the two circles. The common chord AB will then be a common tangent to the two circumferences at their point of contact.

Hence, when two circumferences are tangent to each other, their point of contact is in the straight line joining their centres; and the perpendicular at this point is a common tangent to the two circumferences.

Proposition 12. Theorem.

223. If two circumferences intersect each other, the distance between their centres is less than the sum and greater than the difference of the radii.

Hyp. Let the Os with centres O, O' intersect at A. Join 00', AO, AO'.

To prove 00' < OA + AO', and > OA – AO'.
Proof. In the

OAO'

00' < OA+AO', and 00' > OA – AO'.

Either side of a ▲ < the sum and > the difference of the other two

224. COR. 1. If the distance of the centres of two circles is greater than the sum of their radii, they are wholly exterior to each other.

225. COR. 2. If the distance of the centres of two circles is equal to the sum of the radii, they are tangent externally.

226. COR. 3. If the distance of the centres is less than the sum and greater than the difference of the radii, the circles intersect.

227. COR. 4. If the distance of the centres is equal to the difference of the radii, the circles are tangent internally.

228. Cor. 5. If the distance of the centres is less than the difference of the radii, one circle is wholly within the other.

SCH. If two circles intersect and the radius of either circle drawn to a point of section touches the other circle, the circles intersect orthogonally, i.e., at right angles.

EXERCISES.

1. Through a given point P either inside or outside a given circle whose centre is O, two straight lines PAB, PCD are drawn making equal angles with OP, and cutting the circle in A, B, C, D: prove that AB = CD, and PA PC.

2. P is a point inside a circle whose centre is O: prove that the chord which is at right angles to OP is the shortest chord that can be drawn through P.

Let APB bei to OP, CPD any other chord through P: draw OE to CD, etc.

3. If two circles cut each other, any two parallel straight lines drawn through the points of intersection to cut the circles are equal.

4. Two circles whose centres are A and B intersect at C; through C two chords DCE, FCG are drawn equally inclined to AB and terminated by the circles: show that DEFG.

5. Prove that the two tangents drawn to a circle from an external point are equal and equally inclined to the straight line joining the point to the centre of the circle.

6. A is a point outside a given circle whose centre is 0; with centre A and radius AO a circle is described, and with centre O and radius equal to the diameter of the given circle another circle is discribed cutting the last in B; OB is joined cutting the given circle in E: prove that AE is tangent to the given circle.

THE MEASUREMENT OF ANGLES.

229. To measure a quantity is to find how many times it contains another quantity of the same kind taken as a standard of comparison. This standard is called the unit.

Thus, if we wish to measure a line, we must take a uni of length, and see how many times it is contained in the line to be measured.