geometry – Explanation of Cartesian formula for circumcenter

On Wikipedia there is a Cartesian formula for the circumcenter of a triangle. That is, given points $A$, $B$ and $C$ in $mathbb{R}^2$, find point $U$ such that $d(A,U)=d(B,U)=d(C,U)$. The formula, as stated on Wikipedia, is very algebraic, but I found that it can be rewritten as follows.

Define $X:=(A_x,B_x,C_x)$, $Y:=(A_y,B_y,C_y)$, $V:=(1,1,1)$ and $L:=(|A|^2,|B|^2,|C|^2)$. Then define the matrices $M_D:=(U,X,Y)$, $M_X:=(U,X,L)$ and $M_Y:=(U,L,Y)$. Then we have $$U=(|M_Y|,|M_X|) / (2|M_D|).$$

As I am writing a program, I very much enjoy this elegant form. However, such a nice formula needs a nice explanation if you ask me. Can anyone come up with one? I am hoping for an explanation that does not rely on any of the algebraic properties of the determinant and only uses its geometric interpretation based on measures.