A point has its $x$-coordinate equal to its $y$-coordinate, and it is equidistant from A(2, 8) and B(6, 2). Find the point.

Write the point as (a, a). Equate the squared distances: $(a-2{)}^2+(a-8{)}^2=(a-6{)}^2+(a-2{)}^2$. The $(a-2{)}^2$ terms cancel, leaving $(a-8{)}^2=(a-6{)}^2$, so $-16a+64=-12a+36$, giving $4a=28$ and $a=7$. The point is (7, 7), which is $\sqrt{25+1}=\sqrt{26}$ from each of A and B. (4, 5) is the midpoint of AB, which is equidistant from A and B but does not have equal coordinates.