Question 4
Let \(ABC\) be an acute triangle satisfying
\[\angle ACB\lt \angle ABC\lt \angle ACB+\frac12\angle BAC.\]
Choose \(D\) on segment \(BC\) so that \(\angle ADC=\angle ACB+\tfrac12\angle BAC\). The tangent at \(A\) to the circumcircle of \(ABC\) meets line \(BC\) at \(E\). The internal bisector of \(\angle AEB\) meets \(AD\) at \(G\) and the circle \((ADE)\) again at \(F\ne E\). Put \(H=DF\cap AE\).
(a) Prove that the circles with diameters \(AE,DF,GH\) have a common point.
(b) Choose \(K\) on the external angle-bisector line of \(\angle BAC\) and \(M\) on ray \(AC\) such that \(KB=KD=KM\). Similarly choose \(L\) on that external bisector and \(N\) on ray \(AB\) such that \(LC=LD=LN\). Let \(I\) be the midpoint of \(BC\), and let the circle \((MNI)\) meet \(BC\) again at \(P\ne I\). Prove that \(BM,CN,AP\) are concurrent.
\[\angle ACB\lt \angle ABC\lt \angle ACB+\frac12\angle BAC.\]
Choose \(D\) on segment \(BC\) so that \(\angle ADC=\angle ACB+\tfrac12\angle BAC\). The tangent at \(A\) to the circumcircle of \(ABC\) meets line \(BC\) at \(E\). The internal bisector of \(\angle AEB\) meets \(AD\) at \(G\) and the circle \((ADE)\) again at \(F\ne E\). Put \(H=DF\cap AE\).
(a) Prove that the circles with diameters \(AE,DF,GH\) have a common point.
(b) Choose \(K\) on the external angle-bisector line of \(\angle BAC\) and \(M\) on ray \(AC\) such that \(KB=KD=KM\). Similarly choose \(L\) on that external bisector and \(N\) on ray \(AB\) such that \(LC=LD=LN\). Let \(I\) be the midpoint of \(BC\), and let the circle \((MNI)\) meet \(BC\) again at \(P\ne I\). Prove that \(BM,CN,AP\) are concurrent.