Radiative transfer equation and diffusion theory for photon transport in biological tissue

In mathematics, Casey's theorem, also known as the generalized Ptolemy's theorem, is a theorem in Euclidean geometry named after the Irish mathematician John Casey.

Formulation of the theorem

$t_{12}\cdot t_{34}+t_{41}\cdot t_{23}-t_{13}\cdot t_{24}=0$

Let $\,O$ be a circle of radius $\,R$. Let $\,O_{1},O_{2},O_{3},O_{4}$ be (in that order) four non-intersecting circles that lie inside $\,O$ and tangent to it. Denote by $\,t_{ij}$ the length of the exterior common tangent of the circles $\,O_{i},O_{j}$. Then:

$\,t_{12}\cdot t_{34}+t_{41}\cdot t_{23}=t_{13}\cdot t_{24}.$

Note that in the degenerate case, where all four circles reduce to points, this is exactly Ptolemy's theorem.

Proof

The following proof is due to Zacharias. Denote the radius of circle $\,O_{i}$ by $\,R_{i}$ and its tangency point with the circle $\,O$ by $\,K_{i}$. We will use the notation $\,O,O_{i}$ for the centers of the circles. Note that from Pythagorean theorem,

$\,t_{ij}^{2}={\overline {O_{i}O_{j}}}^{2}-(R_{i}-R_{j})^{2}.$

We will try to express this length in terms of the points $\,K_{i},K_{j}$. By the law of cosines in triangle $\,O_{i}OO_{j}$,

${\overline {O_{i}O_{j}}}^{2}={\overline {OO_{i}}}^{2}+{\overline {OO_{j}}}^{2}-2{\overline {OO_{i}}}\cdot {\overline {OO_{j}}}\cdot \cos \angle O_{i}OO_{j}$

Since the circles $\,O,O_{i}$ tangent to each other:

${\overline {OO_{i}}}=R-R_{i},\,\angle O_{i}OO_{j}=\angle K_{i}OK_{j}$

Let $\,C$ be a point on the circle $\,O$. According to the law of sines in triangle $\,K_{i}CK_{j}$:

${\overline {K_{i}K_{j}}}=2R\cdot \sin \angle K_{i}CK_{j}=2R\cdot \sin {\frac {\angle K_{i}OK_{j}}{2}}$

Therefore,

$\cos \angle K_{i}OK_{j}=1-2\sin ^{2}{\frac {\angle K_{i}OK_{j}}{2}}=1-2\cdot \left({\frac {\overline {K_{i}K_{j}}}{2R}}\right)^{2}=1-{\frac {{\overline {K_{i}K_{j}}}^{2}}{2R^{2}}}$

and substituting these in the formula above:

${\overline {O_{i}O_{j}}}^{2}=(R-R_{i})^{2}+(R-R_{j})^{2}-2(R-R_{i})(R-R_{j})\left(1-{\frac {{\overline {K_{i}K_{j}}}^{2}}{2R^{2}}}\right)$

${\overline {O_{i}O_{j}}}^{2}=(R-R_{i})^{2}+(R-R_{j})^{2}-2(R-R_{i})(R-R_{j})+(R-R_{i})(R-R_{j})\cdot {\frac {{\overline {K_{i}K_{j}}}^{2}}{R^{2}}}$

${\overline {O_{i}O_{j}}}^{2}=((R-R_{i})-(R-R_{j}))^{2}+(R-R_{i})(R-R_{j})\cdot {\frac {{\overline {K_{i}K_{j}}}^{2}}{R^{2}}}$

And finally, the length we seek is

$t_{ij}={\sqrt {{\overline {O_{i}O_{j}}}^{2}-(R_{i}-R_{j})^{2}}}={\frac {{\sqrt {R-R_{i}}}\cdot {\sqrt {R-R_{j}}}\cdot {\overline {K_{i}K_{j}}}}{R}}$

We can now evaluate the left hand side, with the help of the original Ptolemy's theorem applied to the inscribed quadrilateral $\,K_{1}K_{2}K_{3}K_{4}$:

$t_{12}t_{34}+t_{14}t_{23}={\frac {1}{R^{2}}}\cdot {\sqrt {R-R_{1}}}{\sqrt {R-R_{2}}}{\sqrt {R-R_{3}}}{\sqrt {R-R_{4}}}\left({\overline {K_{1}K_{2}}}\cdot {\overline {K_{3}K_{4}}}+{\overline {K_{1}K_{4}}}\cdot {\overline {K_{2}K_{3}}}\right)$

$={\frac {1}{R^{2}}}\cdot {\sqrt {R-R_{1}}}{\sqrt {R-R_{2}}}{\sqrt {R-R_{3}}}{\sqrt {R-R_{4}}}\left({\overline {K_{1}K_{3}}}\cdot {\overline {K_{2}K_{4}}}\right)=t_{13}t_{24}$

Q.E.D.

Further generalizations

It can be seen that the four circles need not lie inside the big circle. In fact, they may be tangent to it from the outside as well. In that case, the following change should be made:

If $\,O_{i},O_{j}$ are both tangent from the same side of $\,O$ (both in or both out), $\,t_{ij}$ is the length of the exterior common tangent.

If $\,O_{i},O_{j}$ are tangent from different sides of $\,O$ (one in and one out), $\,t_{ij}$ is the length of the interior common tangent.

It is also worth noting that the converse of this statement is also true. That is, if equality holds, the circles are tangent.

Applications

Casey's theorem and its converse can be used to prove a variety of statements in Euclidean geometry.

For example, the shortest known proof of Feuerbach's theorem uses the converse theorem.

External links

• 
  
  I had like 17 domains hosted on single account, and never had any special troubles. If you are not happy with the service you will get your money back with in 45 days, that's guaranteed. But the Search Engine utility inside the Hostgator account furnished an instant score for my launched website. Fantastico is unable to install WordPress in a directory which already have any file i.e to install WordPress using Fantastico the destination directory must be empty and it should not have any previous installation files. When you share great information, others will take note. Once your hosting is purchased, you will need to setup your domain name to point to your hosting. Money Back: All accounts of Hostgator come with a 45 day money back guarantee. If you have any queries relating to where by and how to use Hostgator Discount Coupon, you can make contact with us at our site. If you are starting up a website or don't have too much website traffic coming your way, a shared plan is more than enough. Condition you want to take advantage of the worldwide web you prerequisite a HostGator web page, -1 of the most trusted and unfailing web suppliers on the world wide web today. Since, single server is shared by 700 to 800 websites, you cannot expect much speed.
  
  
  
  Hostgator tutorials on how to install Wordpress need not be complicated, especially when you will be dealing with a web hosting service that is friendly for novice webmasters and a blogging platform that is as intuitive as riding a bike. After that you can get Hostgator to host your domain and use the wordpress to do the blogging. Once you start site flipping, trust me you will not be able to stop. I cut my webmaster teeth on Control Panel many years ago, but since had left for other hosting companies with more commercial (cough, cough) interfaces. If you don't like it, you can chalk it up to experience and go on. First, find a good starter template design. When I signed up, I did a search for current "HostGator codes" on the web, which enabled me to receive a one-word entry for a discount. Your posts, comments, and pictures will all be imported into your new WordPress blog.
• Shailesh Shirali: On a generalized Ptolemy Theorem