Parabolas and Tangents

Geometry Level 5

Given \(\triangle ABC\) with \(BC=28, AC=24, AB=20\), two parabolas \(P_B, P_C\) satisfying the following conditions are constructed:

  • \(P_B\) is tangent to segment \(AC\) and rays \(BA,BC\), its focus is denoted \(F_B\)

  • \(P_C\) is tangent to segment \(AB\) and rays \(CA,CB\), its focus is denoted \(F_C\).

  • \(F_BF_C=\frac {35}{\sqrt {3}}\)

  • The axes of symmetry of the parabolas intersect at \(X\), which lies inside the circumcircle of \(ABC\).

Find \(\angle F_BXF_C\) in radians

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