52

EXAMPLES.

EXAMPLES ON CHAPTER II.

1. Describe a circle to pass through two given points, P and P1, and to bisect the circumference of a given circle (centre C, radius CA).

(Draw PC and produce it to D so that PC.CD=AC2. The circle through P, D, P1 fulfils required condition.)

2. Draw two circles cutting orthogonally, and shew by construction that any line through the centre of either cutting both circles is divided harmonically at the points of intersection.

3. Given the base AB of a triangle and the sum of the squares of the sides AC2 + BC2, draw the locus of the vertex.

(A circle, centre at E the middle point of AB, and radius AC2 + BC2

2

AE)

4. Draw two circles (centres A and B) cutting orthogonally, and draw their common chord meeting AB in C. Draw DE a chord of the first circle passing through B, and shew that a circle can be described through ADEC.

5. The centre A of a circle lies on another circle which cuts the former in B, C; AD is a chord of the latter circle meeting BC in E, shew that the polar of D with respect to the first circle passes through E.

6. At two fixed points A, B are drawn AC, BD at right angles to AB and on the same side of it, and of such magnitude that the rectangle AC, BD is equal to the square on AB: prove that the circles whose diameters are AC, BD will touch each other, and that their point of contact will lie on a fixed circle.

(The circle on AB as diameter.)

7. With three given points A, B, C not lying in one straight line as centres describe three circles which shall have three common tangents.

(Bisect the angle BAC by AD meeting BC in D,

then ED, DF, FE will be the required common tangents.)

The question is obviously, given the centres of the escribed circles of a triangle, to draw the triangle.

8. A and B are two given points on the same side of a given straight line CD, which AB meets in C. Determine the points on CD on each side of C at which AB subtends a greater angle than at any other point on the same side.

(The points of contact of circles through A and B, and touching CD. Prob. 21.)

9. A and B are two given points within a circle; and AB is drawn and produced both ways so as to divide the whole circumference into two arcs. Determine the point in each of these arcs at which 4B subtends the greatest angle.

(The points of contact of circles through A and B touching the given circle. Prob. 27.)

10. Shew by construction that the circle which passes through the middle points of the sides of any triangle ABC will pass through the feet of the perpendiculars from A, B, C on the opposite sides, and if O be the intersection of these perpendiculars, will also pass through the middle points of OA, OB, OC. Shew also that it will touch the inscribed and escribed circles of the triangle, and that its radius is half that of the circumscribing circle.

(The circle is called the nine point circle.)

11. Given four points ABCD in a straight line taken in order. Shew that the locus of the point P moving so that the angle APB = the angle CPD, is a circle which may be constructed in the following manner. Let AB be less than CD, and take b between C and D so that bD = AB. The centre is on the given straight line at a distance from A, such that

AO AC: AB : Cb,

:

and the radius (r) is such that

p2 = OB, OC = OA, OD.

12. Find the locus of a point such that the area of the triangle whose angular points are the feet of the perpendiculars from it on the three sides of a given triangle, has a constant area.

[It is a circle of radius p, concentric with the circle circum

scribing the given triangle; and p is determined from the equation

where R is the radius of circumscribing circle, k is the given constant area and A is the area of the given triangle. If 4k<A, is given by the equation

ρ

(Salmon's Conic Sections, Chap. IX.)]

As a numerical example, draw any triangle ABC, and take p: R:: √√7: 1,

13. Given on a straight line four points in the order P, A, B, Q; describe a circle passing through A and B such that tangents drawn to it from P and Q may be parallel.

[With centres P and Q and radii √PA, PB, QA, QB respectively describe two circles. A circle passing through A and B, and through the points of contact of a common tangent to these circles will be the one required.]

14. Given a fixed circle and an external point 0. Draw the tangent at any point P of the circle and complete the rectangle which has OP for side and the tangent for diagonal. Shew that the angular point opposite O will lie on the polar of 0.

15. From the obtuse angle A of a triangle ABC draw a line meeting the base in D so that AD shall be a mean proportional between the segments of the base.

[Find O the centre of the circle circumscribing ABC. On AO as diameter describe a circle cutting the base in D, the required point.] 16. Find on a given line AB a point A such that its polar with respect to a given circle shall pass through a given point C.

[Find P the pole of AB, then the pole of CP will lie on AB i.e. will be the required point A.]

17. Given a point A, a line through it AB, and a circle centre C; draw a triangle APB which shall be self-conjugate with respect to the circle (p. 32).

Take P the pole of the given line and from C draw CB perpendicular to AP meeting AB in B, APB will be the required triangle; for since B is on the polar of P the polar of B will pass through P, and is perpendicular to CB, i. e. is the line AP.

18. Given a triangle APB obtuse-angled at P, to draw the circle with respect to which the triangle shall be self-conjugate.

The centre (C) of the circle must evidently be the intersection of the perpendiculars from the angular points on the opposite sides. Let the perpendicular from P on AB meet it in D. The radius of the circle will be a mean proportional between CP and CD.

19. Given a circle, describe a triangle which shall be selfconjugate with respect thereto, and with its sides parallel to those of a given triangle abp, obtuse-angled at p.

Through C the centre of the given circle draw CA perpendicular to bp, CB perpendicular to ap and CM perpendicular to ab. The vertices of the required triangle will lie, one on each of these lines. Through any point m on CM draw dme perpendicular to CM meeting CA in d and CB in e, and through d draw df perpendicular to CB and Bf perpendicular to CA; f will necessarily lie on CM.

If D is the point on C'm through which the side of the required triangle perpendicular to Cm passes :--

where r is the radius of the given circle, i.e. CD is a mean proportional between r and a length 7 determined by taking a fourth proportional to Cf, Cm, and r; for if

CHAPTER III.

THE PARABOLA.

IF a line be drawn through the centre of a given circle perpendicular to the plane of the circle, the surface generated by a straight line which passes through a fixed point on the first line and moves round the circumference of the circle is called a right circular cone. It will be shewn in Chap. IX. that the intersection of this surface with any plane must be one or other of the following:-a point, a pair of straight lines, a circle, a parabola, an ellipse or an hyperbola. The construction of these last three curves from their definition as the sections of a cone seems à priori to be the natural way of treating the subject; but the fact is they are more easily constructed from some of their known plane properties, and therefore, deferring the consideration of them as lying on the surface of a solid, each will at first be defined as the locus of a point moving in a plane so that its distance from a fixed point is always in a constant ratio to its distance from a fixed line, both point and line being in the plane of motion.

The fixed point is called the focus, and the fixed line the directrix.

In the parabola the ratio is one of equality, i. e. the distance from the fixed point is always equal to the distance from the fixed line.

In the ellipse the ratio is one of less inequality, i. e. the distance from the fixed point is always less than the distance from the fixed line.

In the hyperbola the ratio is one of greater inequality, i. e. the