The polar of a point P with respect to a second-order non-degenerate curve is the set of points N harmonically conjugate with a point P with respect to the points M 1 and M 2 of the intersection of the second-order curve by secants passing through the point P [1] .
Polar is a straight line. Point P is called the polar pole . Every non-degenerate second-order line defines a bijection of the points of the projective plane and the set of its lines — polarization or polar transformation .
Content
Properties
- If the point P lies “outside” the second-order line (that is, two tangents to the line can be drawn through the point P ), then the polar passes through the points of tangency of this line with the lines drawn through the point P.
- If the point P lies on a second-order curve, then the polar is a straight line tangent to the given curve at this point.
- The polar point P passes through its inversion with respect to the corresponding curve. Moreover, if the polar intersects this curve at two points, then the inversion is the middle of the chord with ends at these points.
- The poles of all points lying on a straight line passing through the center of the corresponding curve are parallel to each other. In the case of a parabola, the center is considered to be infinitely distant, the straight line should be parallel to its axis.
- If the polar of the point P passes through the point Q , then the polar of the point Q passes through the point P.
Trilinear Polar Triangle
If we continue the sides of the chevian triangle of some point and take their intersection points with the corresponding sides, then the obtained intersection points will lie on one straight line, called the trilinear polar of the starting point.
- Orthocentric axis - trilinear polar of the orthocenter
- The trilinear polar of the center of the inscribed circle is the axis of the external bisectors .
- The trilinear poles of the points lying on the described conic intersect at one point (for the circumscribed circle this is the Lemoine point , for the described Steiner ellipse it is the centroid ) .
- Chevian triangle - a triangle whose three vertices are the three bases of the Chevian of the original triangle.
History
The term "polar" was introduced by Gergonn .
Variations and generalizations
The polar (polar plane) of a point with respect to a non-degenerate surface of the second order is defined similarly.
The notion of a polar with respect to a second-order line is generalized to an nth- order line. In this case, a n- polar is mapped to a given point on the plane with respect to a line of order n . The first of these polar is a line of order n -1, the second, which is the polar of a given point relative to the first polar, is of the order n -2, etc., and finally, the ( n -1) -th polar is a straight line.
- The trilinear polar of the point Y , isogonally conjugate to the point X , is called the center line of the point X.
- The concept of the center line of point X was introduced by Clark Kimberling in his articles [2] [3] .
See also
- Trilinear Polar Triangles
- Center line (geometry)
Notes
- ↑ A. Savelov. Wonderful curves. Tomsk: Cr. banner, 1938
- ↑ Kimberling, Clark. Central Points and Central Lines in the Plane of a Triangle (English) // Mathematics Magazine : magazine. - 1994 .-- June ( vol. 67 , no. 3 ). - P. 163-187 . - DOI : 10.2307 / 2690608 .
- ↑ Kimberling, Clark. Triangle Centers and Central Triangles . - Winnipeg, Canada: Utilitas Mathematica Publishing, Inc., 1998 .-- P. 285.
Literature
- Efimov N.V., Higher geometry , 6th ed., M., 1978;
- Postnikov M.M., Analytical geometry , M., 1973