Editing Hyperbola (section)

===Inscribed angles for hyperbolas {{math|1=''y'' = ''a''/(''x'' − ''b'') + ''c''}} and the 3-point-form===
[[File:Hyperbel-pws-s.svg|250px|thumb|Hyperbola: inscribed angle theorem]]
A hyperbola with equation <math>y=\tfrac{a}{x-b}+c,\ a \ne 0 </math> is uniquely determined by three points <math>(x_1,y_1),\;(x_2,y_2),\; (x_3,y_3)</math> with different ''x''- and ''y''-coordinates. A simple way to determine the shape parameters <math>a,b,c</math> uses the ''inscribed angle theorem'' for hyperbolas:

{{block indent |em=1.5 |text=In order to '''measure an angle''' between two lines with equations <math>y=m_1x+d_1, \ y=m_2x + d_2\ ,m_1,m_2 \ne 0</math> in this context one uses the quotient

<math display="block">\frac{m_1}{m_2}\ .</math>}}

Analogous to the [[inscribed angle]] theorem for circles one gets the
{{math theorem
|name= Inscribed angle theorem for hyperbolas<ref>[https://www2.mathematik.tu-darmstadt.de/~ehartmann/circlegeom.pdf E. Hartmann: Lecture Note ''Planar Circle Geometries'', an Introduction to Möbius-, Laguerre- and Minkowski Planes, p. 93]</ref><ref>W. Benz: ''Vorlesungen über Geomerie der Algebren'', [[Springer Science+Business Media|Springer]] (1973)</ref>
|math_statement= For four points <math>P_i = (x_i,y_i),\ i=1,2,3,4,\ x_i\ne x_k, y_i\ne y_k, i\ne k</math> (see diagram) the following statement is true:

The four points are on a hyperbola with equation <math>y = \tfrac{a}{x-b} + c</math> if and only if the angles at <math>P_3</math> and <math>P_4</math> are equal in the sense of the measurement above. That means if <math display="block">\frac{(y_4-y_1)}{(x_4-x_1)}\frac{(x_4-x_2)}{(y_4-y_2)}=\frac{(y_3-y_1)}{(x_3-x_1)}\frac{(x_3-x_2)}{(y_3-y_2)}</math>

The proof can be derived by straightforward calculation. If the points are on a hyperbola, one can assume the hyperbola's equation is {{nowrap|<math>y = a/x</math>.}}
}}

A consequence of the inscribed angle theorem for hyperbolas is the
{{math theorem
|name= 3-point-form of a hyperbola's equation
|math_statement= The equation of the hyperbola determined by 3 points <math>P_i=(x_i,y_i),\ i=1,2,3,\ x_i\ne x_k, y_i\ne y_k, i\ne k</math> is the solution of the equation <math display="block">\frac{({\color{red}y}-y_1)}{({\color{green}x}-x_1)}\frac{({\color{green}x}-x_2)}{({\color{red}y}-y_2)}=\frac{(y_3-y_1)}{(x_3-x_1)}\frac{(x_3-x_2)}{(y_3-y_2)}</math> for <math>{\color{red}y}</math>.}}