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But the line 00' always bisects PQ, and therefore it will ultimately pass through 7' the point of contact.

COR. I. Two circles that touch one another have a common tangent at their point of contact.

For the line at right angles to 00' through T is a tangent to both circles by Th. 7.

COR. 2. If R, r are the radii of two circles, D the distance between their centres, it follows that

(1) When the circles intersect,

R+r> D or R−r<D. (1. 13).

(2) When the circles touch,

R+r=D or R-r=D.

(3) When the circles do not meet,

R+r<D or R-r> D.

In other words, if R, r, D are such that any two of them are greater than the third, the circles will intersect; if two of them are together equal to the third, the circles will touch; and if two of them are together less than the third, the circles will not meet, but be wholly inside or wholly outside one another.

EXERCISES.

I. If a straight line touch the inner of two concentric circles, and be terminated by the outer, prove that it will be bisected at the point of contact.

2. Any two chords which intersect on a diameter and make equal angles with it are equal.

3.

Two circles touch each other externally, and a third circle is described touching both externally. Shew that the difference of the distances of its centre from the centre of the two given circles will be constant.

4. If two circles intersect one another, and circles are drawn to touch both, prove that either the sum or the difference of the distances of their centres from the centres of the fixed circles will be constant, according as they touch (1) one internally and one externally, (2) both internally or both externally.

5. If two circles touch one another, any line through the point of contact will cut off segments from the two circles capable of the same angle.

6. If two circles touch one another, two straight lines through the point of contact will cut off arcs, the chords of which are parallel.

7. Two circles cut one another, and lines are drawn through the points of section and terminated by the circumference, shew that they intercept arcs the chords of which are parallel.

8. Circles whose radii are 67 and 7.8 inches are successively placed so as to have their centres 14, 141⁄2, and 15 inches apart. Shew whether the circles will meet or

touch or not meet one another.

9. What will be the case if the centres are 1 inch, I'I inch, or 12 inches apart?

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PROBLEM 2.

To draw a tangent to a circle from a given point.

There will be two cases.

first, let the given point A be on the circumference. Fast (be the centre,

Construction. Join OA, and draw AT at right angles

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Proof. Then AT is a tangent by Th. 7.
Secondly, let A be outside the circle.

Construction. On OA as diameter describe a circle, cutting the given circle in T and T. Join AT, AT'; these shall be tangents from A.

Proof. For join OT, OT'. Then since ATO is a semicircle, the angle ATO is a right angle (II. 5, 2). That is, AT or AT' is at right angles to the radius to the point where it meets the circumference, and therefore AT and AT' are tangents.

It may easily be proved that AT= AT'.

PROBLEM 3.

To cut from any circle a segment which shall be capabie of a given angle.

Let ABC be the circle, D the given angle.

Construction. Take any point A on the circumference. Draw AT the tangent at A (11. 2); and make an angle TAE at A equal to the angle D (1. 5, Cor. 2).

Then shall AE be the chord of the segment required.

B

G

Proof. For the angle in the segment alternate to TAE is equal to the angle TAE, that is, is equal to D.

PROBLEM 4.

On a given straight line to describe a segment of a circle containing an angle equal to a given angle.

Let AB be the given line, C the given angle.

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Construction. At the point A make an angle BAD equal to the angle C (1. 5, Cor. 2).

Then if a circle be described to touch AD in A, and to pass through B, the segment of that circle alternate to BAD will be the segment required.

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