Hello, squat4speed!
What's the orthocenter of a triangle with coordinates:
The orthocenter is the intersection of the three altitudes of the triangle.
Find the equation of two of the altitudes and locate their intersection.
If you make a sketch, the problem is much simpler. Code:
P 
(1,5)* 
* * 
* *
*  *
*  *
*  *
  *  +    *Q
*  * (7,2)
*  *
*+*
* *
* * 
(1,4)* 
R 

Side is vertical.
Hence, the altitude from is the horizontal line, .[1]
Side QR has slope: .
Hence, the altitude from has slope
Its equation is: . .[2]
The intersection of [1] and [2]: .
Therefore: .
The orthocenter is: .