Two of the vertices of a triangle are (1, 4) and (5, −2), and the centroid of the triangle is at (3, 2). What is the third vertex?

The centroid is the average of the three vertices. For the $x$-coordinates, $(1+5+x)/3=3$ gives $6+x=9$, so $x=3$. For the $y$-coordinates, $(4+(-2)+y)/3=2$ gives $2+y=6$, so $y=4$. The third vertex is (3, 4), and checking, $(1+5+3)/3=3$ and $(4-2+4)/3=2$.