Choose an orthogonal system of coordinates such that the center of the sphere with radius $ R$ is $ O = (0,0,0)$, $ A = (0,0,R)$, $ B = (0,0, - R)$. Clearly nonzero reals $ m,n$ exists such that $ M = (0,m,R)$ and $ N = (n,0,R)$, and line $ MN$ is given by $ 2Rx + nz - nR = 0$, $ 2Ry - mz - mR = 0$. Replace the values of $ x,y$ as a function of $ z$ in the equation of the sphere $ x^2 + y^2 + z^2 = R^2$, obtain a 2nd degree equation in $ z$. Since $ MN$ must be tangent to the sphere, the discriminant must be zero, obtain from here condition $ |mn| = 2R^2$ and solution $ z = - R\frac {m^2 - n^2}{m^2 + n^2 + 4R^2}$ for the point of tangency. Substitute in the equations for $ MN$, use $ |mn| = 2R^2$, and find that the point of tangency satisfies $ x = \frac {n|m|}{|m| + |n|}$, $ y = \frac {m|n|}{|m| + |n|}$, or $ |x| = |y|$. Hence $ T$ is necessarily a point in the meridians that result from the intersection of the surface of the sphere with the planes that bisect the dihedron formed by the planes that contain $ AB$ and $ r$, and $ AB$ and $ s$, except for the poles $ A,B$ themselves. Take any point in these meridians, do the construction backward so all these points are in fact the locus of $ T$.

Let $ \alpha , \beta$ denote the planes determined by $ AB , s$ and $ AB , r$ and let $ X,Y$ be the orthogonal projections of $ T$ on $ \alpha$ and $ \beta.$ Since $ X, Y$ lie on the orthogonal projection of $ MN$ on $ \alpha, \beta,$ we get the similar triangles $ \triangle TXN \sim \triangle MAN$ and $ \triangle TYM \sim \triangle NBM$ $\Longrightarrow$ $ \frac {\overline{TX}}{\overline{MA}} = \frac {\overline{NT}}{\overline{MN}} \ , \ \frac {\overline{TY}}{\overline{NB}} = \frac {\overline{MT}}{\overline{MN}}$ Since obviously $ \overline{NT} = \overline{NB}$ and $ \overline{MA} = \overline{MT},$ we deduce that $ \overline{TX} = \overline{TY}$ This means that $ T$ lies on some of the two angle bisector (planes) $ \gamma$ and $ \gamma'$ determined by the dihedral angle $ \alpha\beta.$ Therefore, loci of $ T$ are two circumferences given from the intersection of the sphere with both planes $ \gamma , \gamma'.$