WW.IMOSHL.2022.G4 en
Let \(ABC\) be an acute-angled triangle with \(AC > AB\), let \(O\) be its circumcentre, and let \(D\) be a point on the segment \(BC\). The line through \(D\) perpendicular to \(BC\) intersects the lines \(AO, AC,\) and \(AB\) at \(W, X,\) and \(Y,\) respectively. The circumcircles of triangles \(AXY\) and \(ABC\) intersect again at \(Z \ne A\).
Prove that if \(W \ne D\) and \(OW = OD,\) then \(DZ\) is tangent to the circle \(AXY.\)
Solution-2
Notice that point \(Z\) is the Miquel-point of lines \(AC, BC, BA\) and \(DY\); then \(B,D,Z,Y\) and \(C,D,X,Y\) are concyclic. Moreover, \(Z\) is the centre of the spiral similarity that maps \(BC\) to \(YX\).
By \(BC \perp YX\), the angle of that similarity is \(90^{\circ}\); hence the circles \(ABCZ\) and \(AXYZ\) are perpendicular, therefore the radius \(OZ\) in circle \(ABCZ\) is tangent to circle \(AXYZ\).
By \(OW = OD\), the triangle \(OWD\) is isosceles, and
\[\angle ZOA = 2 \angle ZBA = 2 \angle ZBY = 2 \angle ZDY = \angle ODW + \angle DWO\]
so \(D\) lies on line \(ZO\) that is tangent to circle \(AXY\).