Alegensis game

I made a game for the computer, or an ipad, or android or anything with a browser.
http://alegensis.appspot.com

I'm pretty happy with the way it turned out. I used the Google Web Tools which lets you write Java code that it somehow turns into javascript and html tailored for each of the major browsers. Then you basically just hit a button in the Eclipse IDE and it automatically sends it to be hosted as a Google App Engine site. I chose the name Alegensis because Ale,Gen,Sis is a solution to the puzzle. Those words work vertically as well making Ags, Lei, Ens. 

Crossing point theorem

I found this:

If you know the endpoints of the line, then A for example is the small lines furthest right x coordinate minus it's left x coordinate. The arrow points to what is to subtract what as for vectors. 

E for another example is the y coordinate of the bottom right vertex minus the y coordinate of the small lines left vertex. 

Proof:

Above is a parameterization of two lines A and B with respect to U and T. 

Here is the general solution as Maple solves it. But I'd like to find a better way to write it. 

The denominator of t above can be rewritten as:, immediately below it is the the expansion of it.

The numerator of t as:

Now for a second consider this big fraction to be called W. I'll be plugging this back in for t in the X(T) equation but also rewrite the result of that using this identity:

Which together gives:

Now the nice thing about writing it this way is every term involves only an x or a y and there is one minus sign inside every parenthesis so these can be thought of as distances. Zy works almost exactly the same but with a couple differences. 

These together with variable renaming give the result. 

Where as I explained the end of the arrow points to the term that is subtracting the other term as it would be for vectors. For example the A term in the final equation came from x(2,t) - x(1,t) which is the beginning and end x coordinate of the first line.