Optic equation

 ( Diophantine Equations ) 

In a bicentric quadrilateral, the inradius , the circumradius , and the distance between the incenter and the circumcenter are related by Fuss' theorem according to $$\backslash frac\{1\}\{(R-x)^2\}+\backslash frac\{1\}\{(R+x)^2\}=\backslash frac\{1\}\{r^2\},$$ and the distances of the incenter from the vertices are related to the inradius according to $$\backslash frac\{1\}\{IA^2\}+\backslash frac\{1\}\{IC^2\}=\backslash frac\{1\}\{IB^2\}+\backslash frac\{1\}\{ID^2\}=\backslash frac\{1\}\{r^2\}.$$

In the crossed ladders problem,Gardner, M. Mathematical Circus: More Puzzles, Games, Paradoxes and Other Mathematical Entertainments from Scientific American. New York: Knopf, 1979, pp. 62–64. two ladders braced at the bottoms of vertical walls cross at the height and lean against the opposite walls at heights of and . We have $\backslash tfrac\{1\}\{h\}=\backslash tfrac\{1\}\{A\}+\backslash tfrac\{1\}\{B\}.$ Moreover, the formula continues to hold if the walls are slanted and all three measurements are made parallel to the walls.

Let be a point on the circumcircle of an equilateral triangle , on the minor arc . Let be the distance from to and be the distance from to . On a line passing through and the far vertex , let be the distance from to the triangle side . ThenPosamentier, Alfred S., and Salkind, Charles T., Challenging Problems in Geometry, Dover Publ., 1996. $\backslash tfrac\{1\}\{a\}+\backslash tfrac\{1\}\{b\}=\backslash tfrac\{1\}\{c\}.$

In a trapezoid, draw a segment parallel to the two parallel sides, passing through the intersection of the diagonals and having endpoints on the non-parallel sides. Then if we denote the lengths of the parallel sides as and and half the length of the segment through the diagonal intersection as , the sum of the reciprocals of and equals the reciprocal of . GoGeometry, [1], Accessed 2012-07-08.

The special case in which the integers whose reciprocals are taken must be appears in two ways in the context of . First, the sum of the reciprocals of the squares of the altitudes from the legs (equivalently, of the squares of the legs themselves) equals the reciprocal of the square of the altitude from the hypotenuse. This holds whether or not the numbers are integers; there is a formula (see here) that generates all integer cases. Second, also in a right triangle the sum of the squared reciprocal of the side of one of the two inscribed squares and the squared reciprocal of the hypotenuse equals the squared reciprocal of the side of the other inscribed square.

The sides of a heptagonal triangle, which shares its vertices with a regular heptagon, satisfy the optic equation.

Similarly, the total inductance of two with inductances connected in parallel is given by: $$\backslash frac\{1\}\{L\_1\}\; +\; \backslash frac\{1\}\{L\_2\}\; =\; \backslash frac\{1\}\{L\_t\},$$

and the total capacitance of two with capacitances connected in series is as follows: $$\backslash frac\{1\}\{C\_1\}\; +\; \backslash frac\{1\}\{C\_2\}\; =\; \backslash frac\{1\}\{C\_t\}.$$

• Erdős–Straus conjecture, a different Diophantine equation involving sums of reciprocals of integers
• Sums of reciprocals
• Parallel