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, 14, 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?

SECTION IV.

PROBLEMS.

PROBLEM I.

Given an arc of a circle, to find the centre of the circle

of which it is an arc.

Let ABC be the arc.

Construction. Draw any two

chords AB, BC, and bisect them. at right angles by straight lines ON, OM, intersecting at O. O shall be the centre required.

M

B

Proof. For NO is the locus of points equidistant from A and B, and therefore AO = BO.

Similarly, MO is the locus of points equidistant from B and C; therefore O is equidistant from A, B and C.

Hence, the circle described with centre O and radius equal to one of these three lines, will pass through the other two, and having three points coinciding with the given circular arc, must coincide with it throughout.

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. Let O be the centre.

Construction. Join OA, and draw AT at right angles.

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. (Th. 5, Cor. 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 capable 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; and make an angle TAE at A equal to the angle D.

Then shall AE be the chord of the segment required.

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.

F

E

Construction. At the point A make an angle BAD equal to the angle C.

and

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

To find the centre of this circle, draw AO at right angles to AD: then AO is the locus of the centres of all circles which touch AD at A.

And bisect AB at right angles by the line EO; then EO is the locus of the centres of circles which pass through A and B.

Therefore O, the point of intersection of these lines, is the centre of the circle required.

With centre O and radius OA or OB describe a circle, which will touch AD at A and pass through B, and therefore the segment AFB contains an angle equal to the angle BAD, that is to the given angle C.

PROBLEM 5.

To draw a common tangent to two given circles.

Let the centres of the circles be O, O'.

Construction. With centre O' and radius equal to the sum or difference of the radii of the given circles, describe a circle, as in the figures.

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