14. AB is a fixed chord in a given circle, and from any point C in the arc ACB, a perpendicular CD is drawn to AB. With C as centre and CD as radius a circle is described, and from A and B tangents are drawn to this circle which meet at P; find the locus of P.

15. A quadrilateral inscribed in a circle has one side fixed, and the opposite side constant; find the locus of the intersection of the other two sides, and of the intersection of the diagonals. 16. Two circles touch a given straight line at two given points, and also touch one another; find the locus of their point of contact.

17. Find the locus of the points from which tangents drawn to a given circle may be perpendicular to each other.

18. Find the locus of the points from which tangents drawn to a

given circle may contain a given angle.

19. Find the locus of the points from which tangents drawn to a given circle may be of a given length.

20. From any point on the Oce of a given circle, secants are drawn

such that the rectangle contained by each secant and its exterior segment is constant; find the locus of the ends of the secants.

21. A is a given point and BC a given straight line; any point P is taken on BC, and AP is joined. Find the locus of a point Q taken on AP such that AP AQ is constant.

22. The hypotenuse of a right-angled triangle is given; find the loci of the corners of the squares described outwardly on the sides of the triangle.

23. A variable chord of a given circle passes through a fixed point, and tangents to the circle are drawn at its extremities; prove that the locus of the intersection of the tangents is a straight line. (This straight line is called the polar of the given fixed point, and the given fixed point is called the pole, with reference to the given circle. See the reference to Desargues on p. 221.)

24. Examine the case when the fixed point is outside the circle.

223

BOOK IV.

DEFINITIONS.

1. Any closed rectilineal figure may be called a polygon. Thus triangles and quadrilaterals are polygons of three and four sides.

Polygons of five sides are called pentagons; of six sides, hexagons; of seven, heptagons; of eight, octagons; of nine, nonagons or enneagons; of ten, decagons; of eleven, undecagons or hendecagons; of twelve, dodecagons; of fifteen, quindecagons or pentedecagons; of twenty, icosagons.

Sometimes a polygon having n sides is called an n-gon.

2. A polygon is said to be regular when all its sides are equal, and all its angles equal.

It is important to observe that the triangle is unique among polygons. For if a triangle have all its sides equal, it must have all its angles equal (I. 5, Cor.); if it have all its angles equal, it must have all its sides equal (I. 6, Cor.)

Polygons with more than three sides may have all their sides equal without having their angles equal; or they may have all their angles equal without having their sides equal. A rhombus and a rectangle are illustrations of the preceding remark.

Hence in order to prove a polygon (other than a triangle) regular, it must be proved to be both equilateral and equiangular.

3. When each of the angular points of a polygon lies on the circumference of a circle, the polygon is inscribed in the circle, or the circle is circumscribed about the polygon.

4. When each of the sides of a polygon touches the circumference of a circle, the polygon is circumscribed about the circle, or the circle is inscribed in the polygon.

5. The diagonals of a polygon are the straight lines which join those vertices of the polygon which are not consecutive.

PROPOSITION 1. PROBLEM.

In a given circle to place a chord equal to a given straight line which is not greater than the diameter of the circle.

B

D

Let D be the given straight line which is not greater than the diameter of the given O ABC:

it is required to place in the O ABC a chord = D.

Draw BC any diameter of the O ABC.

Then if BC =

D, what was required is done.

But if not, BC is greater than D.

with centre C and radius CE, describe the O AEF; join CA.

=

Then CA CE, being radii of the O AEF,

= D.

III. 1

Hyp.

I. 3

Const.

1. How many chords can be placed in the circle equal to the given straight line D?

2. Place a chord in the ✪ ABC equal to the given straight line D, and so that one of its extremities shall be at a given point in the Oce. How many chords can be so placed?

3. About a given chord to circumscribe a circle. How many circles can be so circumscribed? Where will their centres all lie? What limits are there to the lengths of the diameters of all such circles?

4. About a given chord to circumscribe a circle having a given radius. How many circles can be so circumscribed ?

Place a chord in the

ABC equal to the given straight line D,

and so that it shall

5. Pass through a given point within the circle.

PROPOSITION 2. PROBLEM.

In a given circle to inscribe a triangle equiangular to a given

Let ABC be the given circle, and DEF the given triangle: it is required to inscribe in ABC a triangle equiangular to A DEF.

Take any point A on the O of ABC,

and at A draw the tangent GAH.

Make HAC = LE, and GAB = LF;

join BC.

III. 17

I. 23

ABC is the required triangle.

Because the chord AC is drawn from A, the point of

.. remaining

BAC remaining 4 D;

I. 32, Cor. 1

▲ : =

.. A ABC is equiangular to A DEF.

1. Show that there may be innumerable triangles inscribed in the O ABC equiangular to the given ▲ DEF.

2. If the problem were, In a given circle to inscribe a triangle equiangular to a given ▲ DEF, and having one of its vertices at a given point A on the Oce, show that six different positions of the inscribed triangle would be possible.

3. Given a

ABC; inscribe in it an equilateral triangle.

4. Two As ABC, LMN are inscribed in the ABC, each of them equiangular to the ▲ DEF; prove ▲s ABC, LMN equal in all respects.

PROPOSITION III. PROBLEM.

About a given circle to circumscribe a triangle equiangular to a given triangle.

-H

E

F

M

B

Let ABC be the given circle, and DEF the given triangle: it is required to circumscribe about ABC a triangle equiangular to ▲ DEF.

Produce EF both ways to G and H.

Find

the centre of the O ABC,

and draw any radius OB.

III. 1

Make BOA = L DEG, and ▲ BOC = ▲ DFH; I. 23 and at A, B, C, draw tangents to the circle intersecting each other at L, M, N.

LMN is the required triangle.