Find The Length

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Let \(A\) and \(B\) be two distinct points on a circle \(\Gamma\) that are not diametrically opposite. Let \(B'\) be the diametrically opposite point of \(B\) on \(\Gamma.\) Let \(C\) be the point on \(\Gamma\) apart from \(B\) such that \(B'A= AC,\) and let \(C'\) be the diametrically opposite point of \(C\) on \(\Gamma.\) If \(AC'= 5,\) find the distance between \(A\) and \(B.\)

Details and assumptions

  • The diametrically opposite point of \(X\) on a circle \(\omega\) is the unique point \(X'\) on \(\omega\) apart from \(X\) such that \(XX'\) is a diameter of \(\omega.\)
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