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(ii) The polar of a given point with respect to a given circle is the straight line drawn through the inverse of the given point at right angles to the line which joins the given point to the centre and with reference to the polar the given point is called the pole.

Thus in the adjoining figure, if OP. OQ=(radius)2, and if through

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P and Q, LM and HK are drawn perp. to OP; then HK is the polar of the point P, and P is the pole of the st. line HK: also LM is the polar of the point Q, and Q the pole of LM.

It is clear that the polar of an external point must intersect the circle, and that the polar of an internal point must fall without it: also that the polar of a point on the circumference is the tangent at that point.

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The Polar of an external point with

reference to a circle is the chord of contact of

tangents drawn from the given point to the circle.

The following Theorem is known as the Reciprocal Property of Pole and Polar.

2. If A and P are any two points, and if the polar of A with respect to any circle passes through P, then the polar of P must pass through A.

Let BC be the polar of the point A with respect to a circle whose centre is O, and let BC pass through P:

then shall the polar of P pass through A.

Join OP; and from A draw AQ perp. to OP. We shall shew that AQ is the polar of P.

Now since BC is the polar of A, ..the ABP is a rt. angle;

and the

Def. 2, page 360.

AQP is a rt. angle:

Constr.

..the four points A, B, P, Q are concyclic;

.. OQ. OP=OA.OB III. 36.

=(radius)2, for CB is the polar of A:

.. P and Q are inverse points with respect to the given circle. And since AQ is perp. to OP,

.. AQ is the polar of P.

That is, the polar of P passes through A.

P

B

Q. E. D.

A similar proof applies to the case when the given point A is without the circle, and the polar BC cuts it.

3. To prove that the locus of the intersection of tangents drawn to a circle at the extremities of all chords which pass through a given point is the polar of that point.

Let A be the given point within the circle, of which O is the centre.

Let HK be any chord passing through A; and let the tangents at H and K intersect at P:

it is required to prove that the locus of P is the polar of the point A.

I. To shew that P lies on the polar of A.

Join OP cutting HK in Q. Join OA: and in OA produced take the point B,

so that OA. OB= (radius)2. II. 14. Then since A is fixed, B is also fixed. Join PB.

H

B

Then since HK is the chord of contact of tangents from P,

.. the

.. OP. OQ=(radius)2. But OA. OB = (radius)2;

.. OP. OQ=OA.OB:

.. the four points A, B, P, Q are concyclic.
at Q and B together two rt. angles.
But the at Q is a rt. angle;

.. the at B is a rt. angle.
And since the point B is the inverse of A;
.. PB is the polar of A;

that is, the point P lies on the polar of A.

Ex. I. p. 233.
Constr.

III. 22. Constr.

Constr

II. To shew that any point on the polar of A satisfies the given conditions.

Let BC be the polar of A, and let P be any point on it. Draw tangents PH, PK, and let HK be the chord of contact.

Now from Ex. 1, p. 366, we know that the chord of contact HK is the polar of P,

and we also know that the polar of P must pass through A; for P is on BC, the polar of A: Ex. 2, p. 367.

that is, HK passes through A.

.. P is the point of intersection of tangents drawn at the extremities of a chord passing through A.

From I. and II. we conclude that the required locus is the polar of A.

NOTE. If A is without the circle, the theorem demonstrated in Part I. of the above proof still holds good; but the converse theorem in Part II. is not true for all points in BC. For if A is without the circle, the polar BC will intersect it; and no point on that part of the polar which is within the circle can be the point of intersection of tangents.

We now see that

(i) The Polar of an external point with respect to a circle is the chord of contact of tangents drawn from it.

(ii) The Polar of an internal point is the locus of the intersections of tangents drawn at the extremities of all chords which pass through it.

(iii) The Polar of a point on the circumference is the tangent at that point.

The following theorem is known as the Harmonic Property of Pole and Polar.

4. Any straight line drawn through a point is cut harmonically by the point, its polar, and the circumference of the circle.

Let AHB be a circle, P the given point and HK its polar; let Paqb be any straight line drawn through P meeting the polar at q and the Oce of the circle at a and b:

then shall P, a, q, b be a harmonic range.

In the case here considered, P is an external point.

Join P to the centre O, and let PO cut the Oce at A and B: let the polar of P cut the ce at H and K, and PO at Q.

A

Then PH is a tangent to the AHB. From the similar triangles OPH, HPQ,

OP: PHPH : PQ,

.. PQ. PO = PH2
= Pa. Pb.

H

K

.. the points O, Q, a, b are concyclic:

.. the aQA = the

B

Ex. 1, p. 366.

abo

the = the

Oab

I. 5.

OQ6, in the same segment.

And since QH is perp. to AB,

.. the aQH-the bQH.

.. Qq and QP are the internal and external bisectors of the LaQb: .. P, a, q, b is a harmonic range.

Ex. 1, p. 360.

The student should investigate for himself the case when P is an internal point.

Conversely, it may be shewn that if through a fixed point P any secant is drawn cutting the circumference of a given circle at a and b, and if q is the harmonic conjugate at P with respect to a, b; then the

locus of q is the polar of P with respect to the given circle.

[For Examples on Pole and Polar, see p. 370.]

Oslove indirez

DEFINITION.

A triangle so related to a circle that each side is the polar of the opposite vertex is said to be self-conjugate with respect to the circle.

H. E

24

EXAMPLES ON POLE AND POLAR.

1. The straight line which joins any two points is the polar with respect to a given circle of the point of intersection of their polars.

2.

The point of intersection of any two straight lines is the pole of the straight line which joins their poles.

3. Find the locus of the poles of all straight lines which pass through a given point.

4. Find the locus of the poles, with respect to a given circle, of tangents drawn to a concentric circle.

5. If two circles cut one another orthogonally and PQ be any diameter of one of them; shew that the polar of P with regard to the other circle passes through Q.

6. If two circles cut one another orthogonally, the centre of each circle is the pole of their common chord with respect to the other circle.

7. Any two points subtend at the centre of a circle an angle equal to one of the angles formed by the polars of the given points.

8. O is the centre of a given circle, and AB a fixed straight line. P is any point in AB; find the locus of the point inverse to P with respect to the circle.

9.

Given a circle, and a fixed point ○ on its circumference: P is any point on the circle: find the locus of the point inverse to P with respect to any circle whose centre is O.

10. Given two points A and B, and a circle whose centre is O; shew that the rectangle contained by OА and the perpendicular from B on the polar of A is equal to the rectangle contained by OB and the perpendicular from A on the polar of B.

11. Four points A, B, C, D are taken in order on the circumference of a circle; DA, CB intersect at P, AC, BD at Q and BA, CD in R: shew that the triangle PQR is self-conjugate with respect to the circle.

12. Give a linear construction for finding the polar of a given point with respect to a given circle. Hence find a linear construction for drawing a tangent to a circle from an external point.

13. If a triangle is self-conjugate with respect to a circle, the centre of the circle is at the orthocentre of the triangle.

14. The polars, with respect to a given circle, of the four points of a harmonic range form a harmonic pencil: and conversely.

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