Given two points: $A$ and $O$. From $O$, we draw circles with varying radius $r$, and from $A$, we draw tangents to these circles.

(a) Find the maximum area of the isosceles triangle formed by the tangents and the chord of contact.

(b) Find the maximum area of the quadrilateral formed by the tangents and the radii to the points of tangency.
a) If the line connecting the points of tangency $C$ and $D$ intersects $A O=d$ at $B$ and $A B=x$, then

$$
T=A B \cdot B C=x \sqrt{x(d-x)}=\sqrt{x^{3}(d-x)}
$$

$T$ is maximized when (K.M.L.VIII.201.l.)

$$
x:(d-x)=3: 1
$$

from which

$$
x=\frac{3}{4} d
$$

in this case:

$$
r=\sqrt{d \cdot \frac{1}{4} d}=\frac{1}{2} d=\frac{1}{2} A O
$$

b) Let the area of the quadrilateral be $T_{1}$, then:

$$
T_{1}=A O \cdot B C=d \sqrt{x(d-x)}
$$

$T_{1}$ is maximized when

$$
x=d-x
$$

from which

$$
x=\frac{1}{2} d
$$

in this case

$$
r=\frac{d}{2} \sqrt{2}=\frac{1}{2} A O \sqrt{2}
$$

(Pivnyik István, Nyíregyháza.)

The problem was also solved by: Bartók I., Dálnoky Nagy Z., Deutsch E., Deutsch I., Enyedi B., Glück I., Haar A., Hirschfeld Gy., Kertész G., Kürti I., Ligeti P., Losonczy I., Pazsiczky G., Preisich G., Raab R., Rássy P., Riesz K., Riesz M., Schuster Gy., Sonnenfeld J., Schwemmer I., Strobl I., Szücs A., Weisz P.