Circumcenter as origin/Sum of points on altitude/Fact/Proof

Proof

We move ${}U$ to the origin, so that the points ${}A,B,C$ have the same norm. The point in question is just ${}A+B+C$. The vector of the line given by this point and ${}A$ is ${}B+C$. This line runs through ${}A$, and we have

${}\left\langle B+C,B-C\right\rangle =\left\langle B,B\right\rangle -\left\langle C,C\right\rangle \,.$

Due to the equality of the norms, this is ${}0$; therefore, this line is the altitude line through ${}A$.

To fact