If you know the values of a function at the three points of a triangle, how can I determine the values at points in the interior? {details}?

There is a triangle ABC. The coordinates of each of the corners are known. In addition, each corner is given an extra value, which is known. Given any two points, the midpoint between them is given the value that is the average of those two points. My goal is to find a way to calculate the values of points directly, given the coordinate of the point.

Here is an illustration I made to show you what I am talking about. The images to the left show that all of the points on the blue lines should all have the same values. On the right, you can see one method of solving a triangle when the three corners are not the same. However, you can see that this process may take an unlimited number of iterations to solve for any particular point.

How do I find the values of any particular point directly, taking the same amount of time for each point?