Question 4
Let $ABC$ be an equilateral triangle and $M$ any point in its plane. Let $A',B',C'$ be the reflections of $A,B,C$, respectively, in the point $M$.
(a) Prove that there is a unique point $P$ equidistant from $A,B'$, from $B,C'$, and from $C,A'$.
(b) Let $D$ be the midpoint of $AB$. As $M$ varies, with $M\ne D$, let $N=DM\cap AP$. Prove that the circumcircle of triangle $MNP$ passes through a fixed point.
Construction domain for (b): a circumcircle is understood only when $M,N,P$ are three distinct noncollinear points. The printed condition $M\ne D$ alone does not ensure this; the exceptional undefined configurations are explicitly discussed in the solutions.
(a) Prove that there is a unique point $P$ equidistant from $A,B'$, from $B,C'$, and from $C,A'$.
(b) Let $D$ be the midpoint of $AB$. As $M$ varies, with $M\ne D$, let $N=DM\cap AP$. Prove that the circumcircle of triangle $MNP$ passes through a fixed point.
Construction domain for (b): a circumcircle is understood only when $M,N,P$ are three distinct noncollinear points. The printed condition $M\ne D$ alone does not ensure this; the exceptional undefined configurations are explicitly discussed in the solutions.