214. Two circles touch each other internally at point $A$. A tangent to the smaller circle intersects the larger circle at points $B$ and $C$. Find the set of centers of circles inscribed in triangles $A B C$.
214. Let the radii of the given circles be denoted by \( R \) and \( r (R > r) \), the point of tangency of the chord \( BC \) with the smaller circle be \( D \); \( K \) and \( L \) will be the points of intersection of the chords \( AC \) and \( AB \) with the smaller circle, and finally, \( O \) is the center of the circle inscribed in \( \triangle ABC \). Since the angular measures of the arcs \( \breve{A} K \) and \( \breve{A} C \) are the same, we can write \( |A K| = r x, \quad |A C| = R x \); from this, we get \( |D C|^2 = A C \cdot |C K| = (R - r) R \lambda^2 \). Similarly, \( |A B| = R y, |D B|^2 = (R - r) R y^2 \); therefore, \( \frac{|C D|}{|D B|} = \frac{x}{y} = \frac{|A C|}{|A B|} \), i.e., \( A D \) is the bisector of angle \( B A C \). Further, we have

\[
\frac{|A O|}{|O D|} = \frac{|A C|}{|C D|} = \frac{R x}{\sqrt{(R - r) R} x} = \sqrt{\frac{R}{R - r}}
\]

Thus, the set of points in question will be a circle that is internally tangent to the two given circles at the same point \( A \), with a radius of

\[
\rho = r \frac{A O'}{\mid A D_1} = \frac{r \sqrt{R}}{\sqrt{R + \sqrt{R - r}}}
\]