[ Extreme properties. Problems on maximum and minimum.] Concerning circles

Chord $A B$ is seen from the center of a circle of radius $R$ at an angle of $120^{\circ}$. Find the radii of the largest circles inscribed in the segments into which chord $A B$ divides the given circle.
The circle of the largest radius inscribed in the given segment with base $AB$ touches the chord $AB$ at its midpoint. Indeed, let $AB$ be a chord of a circle of radius $R$ with center $O$, and $OC$ be the radius of the circle perpendicular to this chord. Then the point $M$ of intersection of segments $AB$ and $OC$ is the midpoint of $AB$. We will prove that the circle with diameter $MC$ is the desired one. Let a circle of radius $x$ with center $O2$, different from the center $O1$ of the circle with diameter $MC=2r$, be inscribed in the given segment $ACB$ and touch the chord $AB$ at point $K$, and the given circle at point $D$. Then the points $O, O2$, and $D$ lie on the same line. Let this line intersect the chord $AB$ at point $P$, and the circle with center $O2$ at point $Q$. Then

$$
R-2r = OM < OP = OD - DP < R - DQ = R - 2x
$$

from which $x < r$.

If now $r$ is the radius of the largest circle inscribed in the smaller of the two given segments, then

$$
r = \frac{1}{2} CM = \frac{1}{2}(R - OM) = \frac{1}{2}\left(R - \frac{1}{2} OA\right) = \frac{1}{2}\left(R - \frac{1}{2} R\right) = \frac{1}{t} R \text{.}
$$

Similarly, we find that the radius of the largest circle inscribed in the second segment is $\frac{3}{4} R$.

## Answer

$\frac{1}{4} R \stackrel{\lrcorner}{\frac{3}{4}} R$