PROPOSITION XIX

209. To draw a common tangent to two given circles, each of which lies wholly outside of the other.

Let C and C1 be the given circles, O and O, their centres, and suppose the radius of C greater than the radius of C1. It is required to draw a straight line which shall touch both of these circles.

Construction.

With centre O and radius equal to the difference between the radii of C and C1 describe a circle C2.

From O, draw a tangent to C2, the point of contact being T. [Notice that two such tangents can be drawn.]

Join OT, and produce to meet Cat T.

2

From O, draw a radius of C1, viz. O,T1, parallel to OT, and on the same side of the line of centres.

Draw the straight line TT,, which shall be the required common tangent.

Proof. The quadrilateral TTO1T, is a parallelogram. Show why.

The angle OTT is a right angle. Why?

Therefore the straight line TT, is perpendicular to a radius of each of the given circles at points T and T1 of the circles. It is therefore tangent to both circles.

EXERCISES

1. Show that by drawing the other tangent from O1 to the circle C2, a second common tangent to the two given circles could be found.

2. Show that by drawing a circle Cs with centre 0 instead of O, and with radius equal to the sum of the radii of the given circles, and also by drawing the parallel radii on opposite sides of the line of centres, two other common tangents to the given circles could be found.

The two common tangents first drawn are called the direct common tangents, the other two the transverse common tangents.

3. Show that when the two given circles are equal the direct common tangents are parallel. How does the construction differ in this case?

4. Show that when the two given circles are in contact, the centre O lies upon the circle C3 (as drawn to obtain the transverse tangents); and that when the given circles intersect, the centre O lies within the circle C's.

Hence, when the two given circles are in contact, the two transverse common tangents coincide and become the common tangent at the point of contact; and when the two given circles intersect, the transverse common tangents cannot be drawn.

5. If in any two given circles which are in contact there be drawn two parallel diameters, the point of contact and an extremity of each diameter lie in the same straight line.

SUGGESTION.

Draw the line of centres, and join an extremity of each diameter to the point of contact. Show that the two lines so drawn make

equal angles with the line of centres.

6. If two circles have a common point not on the line of centres, the circles must intersect at that point, and also at another point which is situated symmetrically with it relative to the line of centres.

SUGGESTION. The foot of the perpendicular from the common point to the line of centres lies inside of both circles; hence the circles must intersect.

7. Two circles touch each other internally or externally at the point A, and through A two straight lines are drawn cutting one circle in P and R, respectively, and the other circle in A and S. Show that PR is parallel to QS.

8. Two circles touch one another at the point A, and have a common tangent meeting them at the points B and C, respectively. Show that the circle whose diameter is BC passes through A. Show also that if the lines BA and CA are produced to cut the circles again at C' and B', respectively, the lines BB' and CC' will be diameters.

SUGGESTION. If the common tangent at A intersects the common tangent BC at M, MA = MB = MC (Art. 194). Therefore BAC is a right angle.

MISCELLANEOUS EXERCISES

1. All chords of the greater of two given concentric circles which are tangent to the smaller are equal.

2. What is the locus of the centres of circles touching two given straight lines which intersect?

3. Describe a circle of given radius which shall touch two given intersecting straight lines. Show that there are four such circles.

4. A circle is described on the radius of another as diameter. Prove that any chord of the greater circle drawn through their point of contact is bisected by the lesser circle.

5. Through a point of intersection of two given circles draw the greatest possible line-segment which is terminated both ways by the circles.

6. If AB and CD are two equal chords in a circle, prove that of the two pairs of straight lines AD, BC, and AC, BD, one pair are equal and the other parallel.

7. Given two circles and a tangent to each, these being parallel; if the points of contact of the tangents be joined by a straight line, the tangents at the points where this straight line cuts the circles a second time are also parallel.

8. Two radii of a circle are at right angles and when produced are cut by a straight line which touches the circle. Show that the other tangents drawn from the points of intersection with the radii are parallel.

9. If two circles touch each other and a chord be drawn through the point of contact, the tangents at the other points where the chord meets the circles are parallel.

10. From all the points of a circle equal and parallel line-segments are drawn in the same direction. What is the locus of their extremities?

drawn to the

What would

11. From any point within a circle straight lines are circle; show that the locus of their mid-points is a circle. be the locus of the mid-points if the lines were drawn from a point on or outside the circle?

12. Let AB be any diameter of a circle whose centre is O, and let C be any point on this diameter produced. Through C draw any secant cutting the circle at D and E. If the exterior part CD of this secant is equal to a radius of the circle, show that the angle EOA is three times the angle DOB.

13. What is the locus of the mid-points of equal chords of a circle ?

14. If two equal chords of a circle are produced to meet outside the circle, prove that the exterior parts are equal. What is the corresponding property when the chords intersect inside the circle? Notice the principle of continuity.

15. What is the locus of the centres of circles of constant radius which touch a given circle?

16. A straight line is drawn intersecting two concentric circles. Show that the line-segments intercepted between the circles are equal.

17. Circles are described on the sides of a quadrilateral as diameters. Show that the common chord of the circles described on two adjacent sides is parallel to the common chord of the other two circles.

18. If AB and A'B' are two equal line-segments lying in a plane, but not parallel, find a point O such that if the line AO be rotated about it through a certain angle, A will coincide with A' and at the same time B with B'.

19. Three circles touch one another externally (each touching the other two), at the points A, B, C ; the straight lines AB, AC, are produced to meet the circle BC at D and E. Show that DE is a diameter of this circle parallel to the line of centres of the other two circles.

20. If AB is a fixed diameter, and DE an arc of constant length in a given circle, and the lines AE, BD intersect at P, show that the angle APB is constant.

21. Three concurrent straight lines make fixed angles with each other. If they be moved so that two of them constantly pass through fixed points, the third must also pass through a fixed point.

22. A triangle is inscribed in a circle. Show that the sum of the angles contained in the three arcs subtended by the sides is equal to four right angles.

23. If ABC is any triangle and a circle is described through the vertices B and C, cutting the sides BA and CA, at the points P and Q, prove that PQ is parallel to a fixed straight line.

24. If E is a point on one of the diagonals AC of a parallelogram ABCD, and circles are described about DEA and BEC, show that the other point of intersection of these two circles must lie on BD.

25. If through P, any point on one of two circles which intersect at A and B, the straight lines PA and PB are drawn to meet the other circle at Q and R, prove that the arc QR is of constant length, or that the length of QR is independent of the position of the point P.

1. DEFINITIONS.

SUMMARY OF CHAPTER II

(1) Circle, centre, radius, diameter. § 147. See also p. 20.

(2) Arc of a Circle — any portion of a circle terminated by two

(5) Semicircle - the arc of a circle subtended by a diameter. § 154. (6) Segment of a Circle - the figure formed by an arc and its subtending chord. § 156.

(7) Sector of a Circle - the figure formed by an arc and the two radii to its extremities. § 156.

(8) Inscribed Figure

a rectilinear figure is said to be inscribed in a circle when its vertices lie on the circle. § 177.

(9) Circumscribed Figure a rectilinear figure is said to be circumscribed about a circle when its sides are all tangent to the circle. § 196.

(10) Concyclic Points-points which lie on the same circle. § 184. (11) Secant of a Circle - an unlimited straight line which intersects a circle in two points. § 185.

(12) Tangent to a Circle- -a straight line which meets a circle, but which when produced does not cut it. § 186.

(13) Chord of Contact - the straight line joining the points of contact of two tangents. § 194.

(14) Circles in Contact

two circles which have a common tangent at a common point. § 205.

(15) Direct and Transverse Common Tangents. See Ex. 2, p. 133. (16) Principle of Continuity - the principle which asserts that a rela

tion among the parts of a geometrical figure, once true and properly interpreted, remains true when the figure changes continuously from one form to another, subject to the conditions under which it was first described. § 204.

2. PROBLEMS.

(1) To find the centre of a circle which passes through three given points, or to circumscribe a circle about a given triangle. § 151. (2) To inscribe a circle in a given triangle. § 196.

(3) To draw a tangent to a circle from a given point on it. § 187.