Question 1
Let the acute triangle $ABC$ be inscribed in $(O)$. Let $A_1,B_1,C_1$ be the feet of its altitudes from $A,B,C$, respectively, and let $A_2,B_2,C_2$ be their reflections in the midpoints of $BC,CA,AB$, respectively. Let $A_3,B_3,C_3$ be the intersections, other than the named vertices $A,B,C$, of the circumcircles of $AB_2C_2,BC_2A_2,CA_2B_2$, respectively, with $(O)$. Prove that $A_1A_3,B_1B_3,C_1C_3$ are concurrent.
Editorial clarification: the source says “the intersections” without specifying the second intersections. When two circles are tangent at a named vertex, the second intersection must be understood with multiplicity; without this convention the statement is not defined for all acute triangles.
Editorial clarification: the source says “the intersections” without specifying the second intersections. When two circles are tangent at a named vertex, the second intersection must be understood with multiplicity; without this convention the statement is not defined for all acute triangles.