Orthogonal Circles

Geometry Level 3

Two circles \(O\) and \(Q\) are orthogonal: their radii are perpendicular at their intersection point \(P,\) as shown above. Then the line \(AD\) is drawn such that it passes through both centers and intersects the arcs at points \(B\) and \(C.\)

If \(AB = 10\) and \(CD = 24\), what is the length of the red segment \(BC?\)

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