BG-MO 1991 Problem 5

On a unit circle with center $O$ , $AB$ is an arc with the central angle $\alpha<90^\circ$ . Point $H$ is the foot of the perpendicular from $A$ to $OB$ , $T$ is a point on arc $AB$ , and $l$ is the tangent to the circle at $T$ . The line $l$ and the angle $AHB$ form a triangle $\Delta$ .

(a) Prove that the area of $\Delta$ is minimal when $T$ is the midpoint of arc $AB$ . (b) Prove that if $S_\alpha$ is the minimal area of $\Delta$ then the function $\frac{S_\alpha}\alpha$ has a limit when $\alpha\to0$ and find this limit.