Quadrilateral In a Circle

Consider a quadrilateral ABCD inscribed in a circle as shown below.

          

The point O is the center of the circle, and we let a,b,c denote
the angles AOD, DOC, and AOB respectively.  The two diagonals of
the quadrilateral meet at the point I.  The distance from O to I
is denoted by s.  If we define  u = (a-b)/2  and  v = (c-a)/2, we 
have

      AD     cos(v) - sin(u)               ____________________
      --  = -----------------        s =  / sin(u)^2 + sin(v)^2
      BC     cos(u) - sin(v)
 
The diagonals are perpindicular if and only if c+b = PI.  An example
of this circumstance is illustrated below.

          

With perpendicular diagonals, we have c=PI-b, so the above expressions 
reduce to

           AD                           __________________
           --  =  tan(a/2)        s =  / 1 - sin(a) sin(b)
           BC

Of course, the area of the quadrilateral is sin(a)+sin(b).  Also in
the case of perpindicular diagonals, suppose we take these two 
diagonals as the diameters of two new circles.  Each of these circles 
intersects the other diagonal in two points.  These four points of 
intersection form a perfect square, with area 2sin(a)sin(b), as
illustrated below.

          

This is a fairly immediate corrollary of Pythagoras's theorem.  If 
AC is the hypotenuse of a right triangle ABC, then the point B is on 
the circle whose diameter is AC, as shown below.

                B
                 *    *
             */ |  \      *
           * /  |     \     *
          * /   |        \   *
        A */____|___________\* C
          *     D            *
          *                  *
           *                *
             *            *
                 *    *

The conventional statement of "right angle relations" is the one
given by Pythagoras and Euclid, which is 

              |AB|^2  + |BC|^2  =  |AC|^2                     (1)

However, another necessary and sufficient condition for right
angles is
                |AD|*|DC|  =  |BD|^2                          (2)

which is sort of the "hyperbolic version" of Pythagoras's theorem.  
Of course (2) is really just a special case of a more general 
relation.  (See Propositions 35 and 36 of Book III of Euclid's
Elements.)  Suppose we have a circle of radius R and an arbitrary 
point on the plane (inside or outside the circle) called P.  Draw 
any line through the point P, and let S,T denote the points of 
intersection with the circle, as shown below

                            /
                 *    *    /
             *            * S
           *             /  *
          *             /    *
          *            /     * 
          *           /      *
          *          /       *
           *        *P      *
             *     /      *
                 */   *
                 / T

Then we have

                |TP|*|PS| =  R^2 - r^2                     (3)

where r is the distance from P to the center of the circle.  
Obviously if TS is a diameter of the circle then R^2 - r^2 is 
just the squared height of the perpindicular at P, and equation 
(3) reduces to (2).  However, the nice thing about the more general 
form (3) is that it works for any line through the circle, not 
just diameters.  Also, note that for any line through the point 
P the product of the two segments is the same (because the 
distance from P to the center of the circle is unchanged).  This 
is why the right triangles drawn on the two diagonals of the 
original quadrilateral have the same heights, resulting in a 
perfect square.

The same theorem applies even when the point P is outside the
circle, as shown below

     \                      
      *P          *    *    
       \      *            * 
        \   *                *
         \ *                  *
          \ T                 *     |PT|*|PS|  =  r^2 - R^2
          *\                  *
          * \                 *
           * \               *
             *\            *
             S \  *    *
                \  
                 \

where again r is the distance from P to the center of the circle.
In fact, this even works when the line through P does not intersect
the circle.  In that case the points of intersection have complex
coordinates in the plane, and the distances |PT| and |PS| are
complex, but they are conjugates, and their product is still the 
real number  r^2 - R^2.  

Naturally this generalizes to other conics as well.  In fact,
given any quadratic curve or surface centered at point O consisting 
of the locus of points that satyisfy the equation such as

                 K^2  =  A*x^2 + B*y^2 + C*z^2

we can define the "distance function" for any two points P1 and P2 
as
     d(P1,P2)^2  =   A*(x1-x2)^2 + B*(y1-y2)^2 + C*(z1-z2)^2

Now, for any line through an arbitrary point P in the space, let 
T1,T2 denote the points of intersection of this line with the given
curve or surface.  The general form of Pythagoras's theorem can
then be expressed as

           K^2   =   d(P,O)^2   +  d(T1,P)*d(P,T2)           (4)

where the sign of d(u,v) is based on a consistent definition of
polarity on the line from u to v.  Thus the signs of d(u,v) and 
d(v,u) are opposite.  Consequently, if the locus is a circle
and the point P is outside the circle, then d(T1,P) and d(P,T2)
have opposite signs, so the right hand term can be brought over
to the left as a positive magnitude.  Moreover, if T1=T2 then
the line from P is tangent to the circle and equation (4) reduces
to the familiar Pythagorean formula

            K^2  +  d(T1,P)^2   =  d(P,O)^2