Find the equation of the circumcircle of triangle $A_{1} A_{2} A_{3}$ in barycentric coordinates.

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Let $X$ be an arbitrary point, $O$ the center of the circumcircle of the triangle, $\mathbf{e}_{\mathrm{i}}=\overrightarrow{\mathrm{OA}_{i}}$ and $\mathbf{a}=\overrightarrow{X O}$. If the point $X$ has barycentric coordinates $\left(x_{1}: x_{2}: x_{3}\right)$, then $\sum x_{\mathrm{i}}\left(\mathbf{a}+\mathbf{e}_{\mathrm{i}}\right)=\sum x_{\mathrm{i}} \overrightarrow{X A_{i}}=0$, since $X$ is the center of mass of the points $A_{1}, A_{2}, A_{3}$ with masses $x_{1}, x_{2}, x_{3}$. Therefore, ( $\sum x_{\mathrm{i}}$ ) $\mathbf{a}=-\sum x_{\mathrm{i}} \mathbf{e}_{\mathrm{i}}$. The point $X$ lies on the circumcircle of the triangle if and only if $|\mathbf{a}|=X O=R$, where $R$ is the radius of this circle. Thus, the circumcircle of the triangle is given in barycentric coordinates by the equation $R^{2}\left(\sum x_{\mathrm{i}}\right)^{2}$ $=\left(\sum x_{\mathrm{i}} \mathbf{e}_{\mathrm{i}}\right)^{2}$, i.e., $R^{2} \sum x_{\mathrm{i}}{ }^{2}+2 R^{2} \sum_{i<j} x_{\mathrm{i}} x_{\mathrm{j}}=R^{2} \sum x_{\mathrm{i}}^{2}+2 \sum_{i<j} x_{\mathrm{i}} x_{\mathrm{j}}\left(\mathbf{e}_{\mathrm{i}}, \mathbf{e}_{\mathrm{j}}\right)$, since $\left|\mathbf{e}_{\mathrm{i}}\right|=R$. This equation can be rewritten as $\sum_{i<j} x_{i} x_{\mathrm{j}}\left(R^{2}-\left(\mathbf{e}_{\mathrm{i}}, \mathbf{e}_{\mathrm{j}}\right)\right)=0$. Note now that $2\left(R^{2}-\left(\mathbf{e}_{\mathrm{i}}, \mathbf{e}_{\mathrm{j}}\right)\right)=a_{\mathrm{ij}}{ }^{2}$, where $a_{\mathrm{ij}}$ is the length of the side $A_{\mathrm{i}} A_{\mathrm{j}}$. Indeed, $a_{\mathrm{ij}}{ }^{2}=\left|\mathbf{e}_{\mathrm{i}}-\mathbf{e}_{\mathrm{j}}\right|^{2}=\left|\mathbf{e}_{\mathrm{i}}\right|^{2}+\left|\mathbf{e}_{\mathrm{j}}\right|^{2}-2\left(\mathbf{e}_{\mathrm{i}}, \mathbf{e}_{\mathrm{j}}\right)=2\left(R^{2}-\left(\mathbf{e}_{\mathrm{i}}, \mathbf{e}_{\mathrm{j}}\right)\right)$. In the end, we obtain that the circumcircle of the triangle $A_{1} A_{2} A_{3}$ is given in barycentric coordinates by the equation $\sum_{i<j} x_{\mathrm{i}} x_{\mathrm{j}} a_{\mathrm{ij}}{ }^{2}=0$, where $a_{\mathrm{ij}}$ is the length of the side $A_{\mathrm{i}} A_{\mathrm{j}}$.