Kvant Math Problem 699

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Problem

A semicircle with diameter $AB$ is cut by the segment $CD$, perpendicular to $AB$, into two curvilinear triangles $ACD$ and $BCD$, in which circles are inscribed, tangent to $AB$ at points $E$ and $F$ (Fig. 1). Prove that

  1. $|AD|=|AF|$,
  2. $[DF]$ is the bisector of angle $BDC$,
  3. the measure of angle $EDF$ does not depend on the choice of the point $C$ on $AB$.

Fig. 1

Fig. 1

V. A. Senderov

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