Question 3
Let triangle \(ABC\) be inscribed in a fixed circle \((O)\), with \(B,C\) fixed and \(A\) moving on one fixed arc \(BC\). The internal angle bisectors \(AD,BE,CF\) meet at \(I\). The circle through \(D\), tangent to line \(OA\) at \(A\), meets \((O)\) again at \(G\). Lines \(GE,GF\) meet \((O)\) again at \(M,N\), respectively, and \(H=BM\cap CN\).
(a) Prove that \(AH\) passes through a fixed point.
(b) Let \(K,L\) be the second intersections of \(BE,CF\) with \((O)\), and let \(P=AH\cap KL\). Choose \(Q\in EF\) with \(QP=QI\). Let \(J\ne I\) be the second intersection of the line through \(I\) perpendicular to \(IQ\) with \((BIC)\). Prove that the midpoint of \(IJ\) lies on a fixed circle.
As is necessary for the construction, consider positions in which all named circles, lines, and intersections exist and are distinct as used.
(a) Prove that \(AH\) passes through a fixed point.
(b) Let \(K,L\) be the second intersections of \(BE,CF\) with \((O)\), and let \(P=AH\cap KL\). Choose \(Q\in EF\) with \(QP=QI\). Let \(J\ne I\) be the second intersection of the line through \(I\) perpendicular to \(IQ\) with \((BIC)\). Prove that the midpoint of \(IJ\) lies on a fixed circle.
As is necessary for the construction, consider positions in which all named circles, lines, and intersections exist and are distinct as used.