Question 6
Let \(ABC\) be an acute triangle with circumcircle \((O)\). Its \(A\)-excircle \((J)\) touches lines \(BC,CA,AB\) at \(D,E,F\), respectively.
(a) Let \(L\) be the midpoint of \(BC\). The circle with diameter \(LJ\) meets lines \(DE,DF\) again at \(K,H\), respectively. Prove that \((BDK)\) and \((CDH)\) have a common point other than \(D\) on \((J)\).
(b) Suppose \(G=EF\cap BC\), and line \(GJ\) meets \(AB,AC\) at \(M,N\). Choose \(P\) on line \(JB\) and \(Q\) on line \(JC\) so that \(AP\perp AB\) and \(AQ\perp AC\). Put \(T=PM\cap QN\). Let \(S\) be the midpoint of the major arc \(BC\) of \((O)\), and \(I\) the incenter. Prove that \(SI\) meets \(AT\) on \((O)\).
Editorial qualification: all named lines, second intersections and circles must be defined and nondegenerate. Acuteness alone does not ensure this, even in part (a). The solution gives explicit counterexamples to the unrestricted printed domain.
(a) Let \(L\) be the midpoint of \(BC\). The circle with diameter \(LJ\) meets lines \(DE,DF\) again at \(K,H\), respectively. Prove that \((BDK)\) and \((CDH)\) have a common point other than \(D\) on \((J)\).
(b) Suppose \(G=EF\cap BC\), and line \(GJ\) meets \(AB,AC\) at \(M,N\). Choose \(P\) on line \(JB\) and \(Q\) on line \(JC\) so that \(AP\perp AB\) and \(AQ\perp AC\). Put \(T=PM\cap QN\). Let \(S\) be the midpoint of the major arc \(BC\) of \((O)\), and \(I\) the incenter. Prove that \(SI\) meets \(AT\) on \((O)\).
Editorial qualification: all named lines, second intersections and circles must be defined and nondegenerate. Acuteness alone does not ensure this, even in part (a). The solution gives explicit counterexamples to the unrestricted printed domain.