National problem · Geometry

Let \(I\) be the centre of the inscribed circle of a triangle \(ABC\), and let \(A_1\), \(B_1\), \(C_1\) be the feet of the perpendiculars dropped from \(I\) to the sides \(BC\), \(AC\) and \(AB\). The line through the midpoints of the segments \(AC_1\) and \(AB_1\), and the line through the midpoints of the segments \(CB_1\) and \(CA_1\), meet at a point \(D\). Prove that the foot of the perpendicular from \(D\) to the side \(AC\) is the midpoint of that side.

\(B_A, C_A, B_C, A_C\) are the midpoints of \(AB_1, AC_1, CB_1, CA_1\); the two lines through them meet at \(D\), and \(E\) is the foot of the perpendicular from \(D\) to \(AC\). The claim is that \(E\) is the midpoint of \(AC\).