Joachimsthal's Notations
Ferdinand Joachimsthal (18181861) was a German mathematician and educator famous for the high quality of his lectures and the books he wrote. The notations named after him and discussed below serve one of the examples where the language of mathematics is especially auspicious for derivation and memorization of properties of mathematical objects. Joachimsthal's notations have had extended influence beyond the study of second order equations and conic sections, compare for example the work of F. Morley.
A general second degree equation
(1)  Ax^{2} + 2Bxy + Cy^{2} + 2Fx + 2Gy + H = 0 
represents a plane conic, or a conic section, i.e., the intersection of a circular twosided cone with a plane. The equations for ellipses, parabolas, and hyperbolas all can be written in this form. These curves are said to be nondegenerate conics. Nondegenerate conics are obtained when the plane cutting a cone does not pass through its vertex. If the plane does go through the cone's vertex, the intersection may be either two crossing straight lines, a single straight line and even a point. These point sets are said to be degenerate conics. In the following, we shall be only concerned with a nondegenerate case.
The lefthand side in (1) will be conveniently denoted as s:
(2)  s = Ax^{2} + 2Bxy + Cy^{2} + 2Fx + 2Gy + H 
so that the second degree equation (1) acquires a very short form:
(3)  s = 0. 
A point P(x_{1}, y_{1}) may or may not lie on the conic defined by (1) or (3). If it does, we get an identity by substituting
(4)  Ax_{1}^{2} + 2Bx_{1}y_{1} + Cy_{1}^{2} + 2Fx_{1} + 2Gy_{1} + H = 0, 
which has a convenient Joachimsthal's equivalent
(5)  s_{11} = 0. 
For another point P(x_{2}, y_{2}) we similarly define s_{22} and, in general, for points
(6)  s_{ii} = Ax_{i}^{2} + 2Bx_{i}y_{i} + Cy_{i}^{2} + 2Fx_{i} + 2Gy_{i} + H. 
Thus, s_{ii} = 0 means that P(x_{i}, y_{i}) lies on the conic (3),
There is also a mixed notation. For two points P(x_{i}, y_{i}) and
(7)  s_{ij} = Ax_{i}x_{j} + B(x_{i}y_{j} + x_{j}y_{i}) + Cy_{i}y_{j} + F(x_{i} + x_{j}) + G(y_{i} + y_{j}) + H. 
Clearly for P(x_{i}, y_{i}) = P(x_{j}, y_{j}), (7) reduces to (6). An important observation is that s_{ij} is symmetrical in its indices:
(8)  s_{ij} = s_{ji}. 
The last of Joachimsthal's conventions brings the first whiff of an indication as to how useful the notations may be. In s_{ij} both
(9)  s_{i} = Ax_{i}x + B(x_{i}y + xy_{i}) + Cy_{i}y + F(x_{i} + x) + G(y_{i} + y) + H. 
The curious thing about (9) is that, although s_{ij} was probably perceived as a number, s_{i} appears to dependent on "variable" x and y and thus is mostly perceived as a function of these variables. As a function of x and y, (9) is linear, i.e. of first degree, so that
Theorem
Let point P(x_{i}, y_{i}) lie on the conic s = 0. In other words, assume that
Proof
Any point P(x, y) on the line through two distinct points P(x_{1}, y_{1}) and P(x_{2}, y_{2}) is a linear combination of the two points:
(10)  P(x, y) = t·P(x_{1}, x_{1}) + (1  t)·P(x_{2}, x_{2}), 
which is just a parametric equation of the straight line. Substitute (10) into (2). The exercise may be a little tedious but is quite straightforward. The result is a quadratic expression in t:
(11)  s(t) = t^{2}·(s_{11} + s_{22}  2s_{12}) + 2t·(s_{12}  s_{22}) + s_{22}. 
Line (10) and conic (1) will have 0, 1, or 2 common points depending on the number of roots of the quadratic equation
(12) 

The line is tangent to the conic iff the quadratic equation has two equal roots, i.e. when
(13)  s_{12}^{2} = s_{11}·s_{22}. 
This is an interesting identity valid for any line tangent to the conic, with
(14)  s_{12} = 0. 
Now, since this is true for any point
(15)  s_{1} = 0, 
which proves the theorem.
Tangent Pair
If two points P(x_{1}, y_{1}) and P(x_{2}, y_{2}) are such that the line joining them is tangent to a conic s = 0, then as in (13),
(16)  s_{1}^{2} = s_{11}·s. 
The latter is a quadratic equation which may be factorized into the product of two linear equations each representing a tangent to the conic through
Example
Let s = x^{2} + 4y^{2}  25, so that
s_{11} = 25 and s_{1} = 25. 
So that (16) becomes
s = 25, or x^{2} + 4y^{2} = 0. 
Obviously the equation has no real roots (besides
s_{11} = 25 and s_{1} = 5x + 10y  25. 
(16) then becomes
(5x + 10y  25)^{2} = 25·(x^{2} + 4y^{2}  25). 
First, let's simplify this to
(x + 2y  5)^{2} = x^{2} + 4y^{2}  25. 
Second, let's multiply out and simplify by collecting the like terms:
2xy  5x  10y + 25 = 0, 
which is factorized into
(x  5)·(2y  5) = 0. 
Conclusion: here are two tangents from (5, 5/2) to the ellipse:
Poles and Polars With Respect To a Conic
Let P(x_{1}, y_{1}) be a point outside a conic
Then the tangents have the equations (15)
(17)  s_{2} = 0 and s_{3} = 0 
and also meet at P(x_{1}, y_{1}):
(18)  s_{21} = 0 and s_{31} = 0. 
Because of the symmetry of the notations, we have
(19)  s_{12} = 0 and s_{13} = 0, 
which says that points
(20)  s_{1} = 0. 
The latter is uniquely determined by P(x_{1}, y_{1}), which, too, can be retrieved from (20). We define
Thus we see that the pole/polar definitions generalize naturally from the circle to other nondegenerate conics. We now prove La Hire's
Theorem
If point P(x_{1}, y_{1}) lies on the polar of P(x_{2}, y_{2}) with respect to a conic
Proof
Ineed, P(x_{1}, y_{1}) lies on the polar
Beautiful.
References
 D. A. Brannan et al, Geometry, Cambridge University Press, 2002
Poles and Polars
 Poles and Polars
 Brianchon's Theorem
 Complete Quadrilateral
 Harmonic Ratio
 Harmonic Ratio in Complex Domain
 Inversion
 Joachimsthal's Notations
 La Hire's Theorem
 La Hire's Theorem, a Variant
 La Hire's Theorem in Ellipse
 Nobbs' Points, Gergonne Line
 Polar Circle
 Pole and Polar with Respect to a Triangle
 Poles, Polars and Quadrilaterals
 Straight Edge Only Construction of Polar
 Tangents and Diagonals in Cyclic Quadrilateral
 Secant, Tangents and Orthogonality
 Poles, Polars and Orthogonal Circles
 Seven Problems in Equilateral Triangle, Solution to Problem 1
Conics
 Conic Sections
 Conic Sections as Loci of Points
 Construction of Conics from Pascal's Theorem
 Cut the Cone
 Dynamic construction of ellipse and other curves
 Joachimsthal's Notations
 MacLaurin's Construction of Conics
 Newton's Construction of Conics
 Parallel Chords in Conics
 Theorem of Three Tangents to a Conic
 Three Parabolas with Common Directrix
 Butterflies in a Pencil of Conics
 Ellipse
 Parabola
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