POLE AND POLAR
§ 62. We return once again to fig. 21, which we obtained in § 55.
If a fourside be circumscribed about and a fourpoint inscribed in a conic, so that the vertices of the second are the points of contact of the sides of the first, then the triangle formed by the diagonals of the first is the same as that formed by the diagonal points of the other.
Such a triangle will be called a polartriangle of the conic, so that PQR in fig. 21 is a polartriangle. It has the property that on the side p opposite P meet the tangents at A and B, and also those at C and D. From the harmonic properties of fourpoints and foursides it follows further that the points L, M, where it cuts the lines AB and CD, are harmonic conjugates with regard to AB and CD respectively.
If the point P is given, and we draw a line through it, cutting the conic in A and B, then the point Q harmonic conjugate to P with regard to AB, and the point H where the tangents at A and B meet, are determined. But they lie both on p, and therefore this line is determined. If we now draw a second line through P, cutting the conic in C and D, then the point M harmonic conjugate to P with regard to CD, and the point G where the tangents at C and D meet, must also lie on p. As the first line through P already determines p, the second may be any line through P. Now every two lines through P determine a fourpoint ABCD on the conic, and therefore a polartriangle which has one vertex at P and its opposite side at p. This result, together with its reciprocal, gives the theorems
All polartriangles which have one vertex in common have also the opposite side in common.
All polartriangles which have one side in common have also the opposite vertex in common.
§ 63. To any point Pin the plane of, but not on, a conic corresponds thus one line p as the side opposite to P in all polartriangles which have one vertex at P, and reciprocally to every line'p corresponds one point P as thevertex opposite to p in all triangles which have p as one side.
We call the line p the polar of P, and the point P the pole of the line p with regard to the conic.
If a point lies on the conic, we call the tangent at that point its polar; and reciprocally we call the point of contact the pole of tangent.
§ 64. From these definitions and former results follow
The polar of any point P not The pole of any line p not a
on the conic is a line p, which has tangent to the conic is a point
the following properties:— P, which has the following pro
perties:
I. On every line through P I. Of all lines through a point which cuts the conic, the polar on p from which two tangents of P contains the harmonic con may be drawn to the conic, the jugate of P with regard to those pole P contains the line which is
points on the conic. harmonic conjugate to p, with
regard to the two tangents.
2. If tangents can be drawn 2. If p cuts the conic, the
from P, their points of contact lie tangents at the intersections
on p. meet at P.
s3. Tangents drawn at the 3. The point of contact of points where any line through P tangents drawn from any point cuts the conic meet on p; and on p to the conic lie in a line with
conversely, P; and conversely,
4. If from any point on p, 4. Tangents drawn at points tangents be drawn, their points where any line through P cuts the of contact will lie in a line with P. conic meet on p.
5. Any fourpoint on the conic 5. Any fourside circumscribed which has one diagonal point at about a conic which has one P has the other two lying on p. diagonal on p has the other two
meeting at P.
The truth of 2 follows from I. If T be a point where p cuts the conic, then one of the points where PT cuts the conic, and which are harmonic conjugates with regard to PT, coincides with T; hence the other does—that is, PT touches the curve at T.
That 4 is true follows thus: If we draw from a point H on the polar one tangent a to the conic, join its point of contact A to the pole P, determine the second point of intersection B of this line with the conic, and draw the tangent at B, it will pass through H, and will therefore be the second tangent which may be drawn from H to the curve.
§ 65. The second property of the polar or pole gives rise to the theorem
From a point in the plane of a A line in the plane of a conic conic, two, one or no tangents has two, one or no points in may be drawn to the conic, common with the conic, accordaccording as its polar has two, ing as two, one or no tangents one, or no points in common with can be drawn from its pole to the
the curve. conic.
Of any point in the plane of a conic we say that it was without, on or within the curve according as two, one or no tangents to the curve pass through it. The points on the conic separate those within the conic from those without. That this is true for a circle is known from elementary geometry. That it also holds for other conics follows from the fact that every conic may be considered as the projection of a circle, which will be proved later on.
The fifth property of pole and polar stated in § 64 shows how to find the polar of any point and the pole of any line by aid of the straightedge only. Practically it is often convenient to draw three secants through the pole, and to determine only one of the diagonal points for two of the fourpoints formed by pairs of these lines and the conic (fig. 22).
These constructions also solve the problem
From a point without a conic, to draw the two tangents to the conic by aid of the straightedge only.
For we need only draw the polar of the point in order to find the points of contact.
§ 66. The property of a polartriangle may now be stated thus—In a polartriangle each side is the polar of the opposite vertex, and each vertex is the pole of the opposite side.
If P is one vertex of a polartriangle, then the other vertices, Q and R, lie on the polar p of P. One of these vertices we may choose arbitrarily. For if from
any point Q_ on the polar B a secant be drawn cutting the conic in A and D (fig. 23), and if the lines joining these points to P cut the conic again at B and C, then the line BC will pass through Q. Hence P and Q are two of the vertices on the polartriangle which is determined by the fourpoint ABCD. The third vertex R lies also on the line p. It follows, therefore, also
If Q is a point on the polar of P, then P is a point on the polar of Q; and reciprocally,
If q is a line through the pole of p, then p is a line through the pole of q.
This is a very important theorem. It may also be stated thus
If a point moves along a line describing a row, its polar turns about the pole of the line describing a pencil.
This pencil is projective to the row, so that the crossratio of four poles in a row equals the crossratio of its four polars, which pass through the pole of the row.
To prove the last part, let us suppose that P, A and B in fig. 23 remain fixed, whilst Q moves along the polar p of P. This will make CD turn about P and move R along p, whilst QD and RD describe projective pencils about A and B. Hence Qand R describe projective rows, and hence PR, which is the polar of Q, describes a pencil projective to either.
§ 67. Two points, of which one, and therefore' each, lies on the polar of the other, are said to be conjugate with regard to the conic; and two lines, of which one, and therefore each, passes through the pole of the other, are said to be conjugate with regard to the conic. Hence all points conjugate to a point P lie on the polar of P; all lines conjugate'to a line p pass through the pole of p.
If the line joining two conjugate poles cuts the conic, then the poles are harmonic conjugates with regard to the points of intersection; hence one lies within the other without the conic, and all points conjugate to a point within a conic lie without it.
Of a polartriangle any two vertices are conjugate poles, any two sides conjugate lines. If, therefore, one side cuts a conic, then one of the two vertices which lie on this side is within and the other ,without the conic. The vertex opposite this side lies also without, for it is the pole of a line which cuts the curve. In this case therefore one vertex lies within, the other two without. If, on the other hand, we begin with a side which does not cut the conic, then its pole lies within and the other vertices without. Hence
Every polartriangle has one and only one vertex within the conic.
We add, without a proof, the theorem
The four points in which a conic is cut by two conjugate polars are four harmonic points in the conic.
§ 68. If two conics intersect in four points (they cannot have more points in common, § 52), there exists one and only one Id
fourpoint which is inscribed in both, and therefore one polartriangle common to both.
Theorem.—Two conics which intersect in four points have always one and only one common polartriangle; and reciprocally,
Two conics which have four common tangents have always one and only one common polartriangle.
End of Article: POLE AND 

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