The Journal of Online Mathematics and Its Applications, Volume 8 (2008)
The Most Marvelous Theorem in Mathematics, Dan Kalman

## Mathematical Background Outline

1. Polynomials
2. Complex Numbers
1. Basic definitions and properties
2. Polar form and geometric interpretations: writing any complex number as x + iy = reiq; multiplying and adding complex numbers geometrically
3. Geometric condition for zw to be a negative real, where z and w are complex numbers
3. Linear and geometric transformations of the plane
4. Ellipses
1. Overview concepts of ellipses and conic sections
2. Definition of ellipse as a deformed circle
3. Standard equation for an ellipse:
4. Equivalent definition: set of points the sum of whose distances from two given points is fixed
5. Linear transformations take ellipses to ellipses
6. First uniqueness property: Given two points E and F and a point P not on the segment between E and F, there is a unique ellipse through P having foci at E and F
7. Second uniqueness property: Given two points E and F and a line L not intersecting the segment between E and F, there is a unique ellipse tangent to L having foci at E and F
8. Optical Property: At any point P of an ellipse, the lines to P from the foci make equal acute angles with the tangent to the ellipse at P.
9. Extended Optical Property: At any point P outside of an ellipse, the lines to P from the foci make equal acute angles with the two tangent lines to the ellipse from the point P.
5. Every triangle has a unique inscribed ellipse, tangent at midpoints