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 four-point ABCD on the conic, and therefore a polar-triangle which has one vertex at P and its opposite side at p. This result, together with its reciprocal, gives the theorems—
All polar-triangles which have one vertex in common have also the opposite side in common.
All polar-triangles which have one side in common have also the opposite vertex in common.
§ 63. To any point P in the plane of, but not on, a conic corresponds thus one line p as the side opposite to P in all polar-triangles which have one vertex at P, and reciprocally to every line p corresponds one point P as the vertex 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 on the conic is a line p, which has the following properties:— | The pole of any line p not a tangent to the conic is a point P, which has the following properties:— |
| 1. On every line through P which cuts the conic, the polar of P contains the harmonic conjugate of P with regard to those points on the conic. | 1. Of all lines through a point on p from which two tangents may be drawn to the conic, the pole P contains the line which is harmonic conjugate to p, with regard to the two tangents. |
| 2. If tangents can be drawn from P, their points of contact lie on p. | 2. If p cuts the conic, the tangents at the intersections meet at P. |
| 3. Tangents drawn at the points where any line through P cuts the conic meet on p; and conversely, | 3. The point of contact of tangents drawn from any point on p to the conic lie in a line with P; and conversely, |
| 4. If from any point on p, tangents be drawn, their points of contact will lie in a line with P. | 4. Tangents drawn at points where any line through P cuts the conic meet on p. |
| 5. Any four-point on the conic which has one diagonal point at P has the other two lying on p. | 5. Any four-side circumscribed about a conic which has one diagonal on p has the other two meeting at P. |
The truth of 2 follows from 1. 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.