Two circles intersect at two points \(B\) and \(C\). Through \(B\), two-line segments \(ABD\) and \(XBY\) are drawn to intersect the circles at \(A, D\), and \(X, Y\) respectively (see given figure). Check that \(∠ACX=∠YCD\).
Proof:
Join chords \(AX\) and \(DY\).
For chord \(AX\), \(∠XBA =\) () ... (1)
For chord \(DY\), \(∠DBY =\) () ... (2)
\(ABD\) and \(XBY\) are line segments intersecting at \(B\).
Therefore, \(∠XBA =\) () ... (3)
From equations (1), (2), and (3),
We obtain \(∠ACX = \)
Hence, proved.