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triangle based on arc, area minimization

Source: Bulgaria 1991 P5

June 3, 2021
geometry

Problem Statement

On a unit circle with center OO, ABAB is an arc with the central angle α<90∘\alpha<90^\circ. Point HH is the foot of the perpendicular from AA to OBOB, TT is a point on arc ABAB, and ll is the tangent to the circle at TT. The line ll and the angle AHBAHB form a triangle Δ\Delta.
(a) Prove that the area of Δ\Delta is minimal when TT is the midpoint of arc ABAB. (b) Prove that if SαS_\alpha is the minimal area of Δ\Delta then the function Sαα\frac{S_\alpha}\alpha has a limit when α→0\alpha\to0 and find this limit.
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