Geometry Stumper

[Albert Chan]

Hi, Thomas Puettmann,

I don't know about computers replacing human to do "math".
But, it allowed you to do things you would never dare checking.

For example, how about proving a straight line by adding up distance ?
Warning: do not do this by hand ! Even CAS have a tough time ...

In[1]:= pointE = {x, y} /. Solve[y == ae*x == be*(x - 1), {x, y}] // First;
In[2]:= pointD = pointE /. {ae -> -1/ae, be -> -1/be} // Simplify;
In[3]:= pointC = {k, 1};

Code:

In[4]:= distance[pt_] := Sqrt[pt[[1]]^2 + pt[[2]]^2];
In[5]:= Solve[distance[pointD - pointC] + distance[pointC - pointE] == distance[pointD - pointE], k]
                  2          2        2
                ae  + be + ae  be - be
Out[5]= {{k -> --------------------------}}
                            2           2
               -ae + be + ae  be - ae be

In[6]:= % /. {ae^2 -> 1 - 2*k*ae, be^2 -> 1 - 2*(k-1)*be} // Simplify
Out[6]= {{k -> k}}

Solving for value of k so DCE is straight, we got {k -> k}
That means DCE is always straight