The points $(1, 7), (13, 16)$ and $(5, k)$, where $k$ is an integer, are vertices of a non-degenerate triangle. What is the sum of the values of $k$ for which the area of the triangle is a minimum?
See the image :
The two points that minimize the area are ( 5, 9) and (5, 11)
The sum of all possible k's = 9 + 11 = 20
+184 
