5.135*. Let vertices $B$ and $C$ of a triangle be fixed, and vertex $A$ move such that the Brocard angle $\varphi$ of triangle $A B C$ remains constant. Then point $A$ moves along a circle of radius $(a / 2) \sqrt{\operatorname{ctg}^{2} \varphi-3}$, where $a=B C$ (Neuberg circle).
5.135. According to problem 12.44, a)

$$
\operatorname{ctg} \varphi=\frac{a^{2}+b^{2}+c^{2}}{4 S}
$$

where $S$ is the area of the triangle. Thus, for a triangle with vertices at points with coordinates $( \pm a / 2,0)$ and $(x, y)$, the Brocard angle $\varphi$ is determined by the equation

$$
\operatorname{ctg} \varphi=\frac{a^{2}+(a / 2+x)^{2}+y^{2}+(-a / 2+x)^{2}+y^{2}}{2 a y}
$$

i.e., $2 x^{2}+2 y^{2}+3 a^{2} / 2=2 a y \operatorname{ctg} \varphi$. The last equation defines a circle of radius $(a / 2) \sqrt{\operatorname{ctg}^{2} \varphi-3}$ with center $(0,(a / 2) \operatorname{ctg} \varphi)$.