Proposition 37. Problem.

275. On a given straight line, to describe a segment which shall contain a given angle.

With O as a centre, and OA as a radius, describe the Oce AHBF.

The segment AHB is the segment required.

Proof. Join OB and BH. Since O is in the AB, it is equally distant from the pts. A, B.

bisector of

(66)

.. a described with centre O and radius OA must pass

through B.

Also, since AD is to AH at its extremity,

NOTE. In the particular case when the given angle C is a right angle, the seg

ment required will be the semicircle described on the given st. line AB.

let

Proposition 38. Problem.

276. To draw a common tangent to two given circles. Given, the Os AR, BS, and

AR > BS.

(1) Required, to draw an exterior common tangent to the two Os.

Cons. With centre A and radius the difference of the radii of the two Os, describe a O.

From B draw BC tangent to this O.

S

R

E

(268)

Join AC, and produce it to meet the Oce of the given

O in D.

Draw BE || to AD, and join DE.

DE is a common tangent to the two given Os.

A tang, to a at any pt. is to the radius at that pt. (210).

.. BCDE is a rectangle.

.. ¿DE = a rt. 2. .. DE is a tangent to both OS.

(77)

Q. E. F.

AR,

Since two tangents can be drawn from B to the therefore two common tangents may always be drawn to the given Os. These are called the direct common tangents.

When the given Os are external to each other and do not intersect, two more common tangents may be drawn, called the transverse common tangents.

(2) Required, to draw the transverse pair of common tangents.

Cons. and proof. With centre A and radius= the sum of the radii of the two

Os, describe a O, and complete the construction, and proof, as in (1).

EXERCISES.

THEOREMS.

1. If a straight line cut two concentric circles, the parts of it intercepted between the two circumferences are equal. 2. If one circle touch another internally at P, prove that the straight line joining the extremities of two parallel diameters of the circles, towards the same parts, passes through P.

3. In Ex. 2, if a chord AB of the larger circle touches the smaller one at C, prove that PC bisects the angle APB.

4. If two circles touch externally at P, prove that the straight line joining the extremities of two parallel diameters towards opposite parts, passes through P.

5. Two circles with centres A and B touch each other externally, and both of them touch another circle with centre O internally show that the perimeter of the triangle AOB is equal to the diameter of the third circle.

6. In two concentric circles any chord of the outer circle which touches the inner, is bisected at the point of contact. 7. If three circles touch one another externally in P, Q, R, and the chords PQ, PR of two of the circles be produced to meet the third circle again in S, T, prove that ST is a diameter.

8. Points P, Q, R on a circle, whose centre is 0, are joined; OM, ON are drawn perpendicular to PQ, PR respectively join MN, and show that if the angle OMN

is greater than ONM, then the angle PRQ is greater than PQR.

9. A circle is described on the radius of another circle as diameter, and two chords of the larger circle are drawn, one through the centre of the less at right angles to the common diameter, and the other at right angles to the first through the point where it cuts the less circle. Show that these two chords have their greater segments equal to each other and their less segments equal to each other.

10. O is the centre of a circle, P is any point in its circumference, PN a perpendicular on a fixed diameter: show that the straight line which bisects the angle OPN always passes through one or the other of two fixed points on the circumference.

11. Two tangents are drawn to a circle at the opposite extremities of a diameter, and intercept from a third tangent a portion AB: if C be the centre of the circle show that ACB is a right angle.

12. A straight line touches a circle at A, and from any point P, in the tangent, PB is drawn meeting the circle at B so that PB is equal to PA: prove that PB touches the circle.

13. OC is drawn from the centre O of a circle perpendicular to a chord AB: prove that the tangents at A, B intersect in OC produced.

14. TA, TB are tangents to a circle, whose centre is 0; from a point P on the circumference a tangent is drawn cutting TA, TB or those produced in C, D: prove that the angle COD is half the angle AOB.

15. AB is the diameter and C the centre of a semicircle: show that O the centre of any circle inscribed in the semicircle is equidistant from C and from the tangent to the semicircle parallel to AB.

16. If from any point without a circle straight lines be drawn touching it, the angle contained by the tangents is double the angle contained by the straight line joining the

points of contact and the diameter drawn through one of them.

17. C is the centre of a given circle, CA a radius, B a point on a radius at right angles to CA; join AB and produce it to meet the circle again at D, and let the tangent at D meet CB produced at E: show that BDE is an isosceles triangle.

18. Let the diameter BA of a circle be produced to P, so that AP equals the radius; through A draw the tangent AED, and from P draw PEC touching the circle at C and meeting the former tangent at E; join BC and produce it to meet AED at D: then will the triangle DEC be equilateral.

Let O be the centre of the given . Produce OC to a pt. F so that CF = CO: compare as PCO, PCF, etc.

19. APB is a fixed chord passing through P, a point of intersection of two circles AQP, PBR; and QPR is any other chord of the circles passing through P: show that AQ and RB when produced meet at a constant angle.

20. Two circles whose centres are A and B touch externally at C; the common tangent at C meets another common tangent DE at F: prove that (1) CF, DF, FE are equal; (2) each of the angles AFB, DCE is a right angle; (3) DE touches the circle described on AB as diameter.

21. The diagonals AC, BD of a quadrilateral ABCD inscribed in a circle intersect at right angles at P: prove that the straight line drawn from P to the middle point of one of the sides of the quadrilateral is perpendicular to the opposite sides.

Bisect AB in E, produce EP to meet CD in F, etc.

22. If a side of a quadrilateral inscribed in a circle be produced, the exterior angle is equal to the interior and opposite angle: and conversely, if the exterior angle of a quadrilateral made by any side and the adjacent side produced be equal to the interior and opposite angle, a circle can be described about the quadrilateral.