City problem · Geometry

In a triangle \(ABC\) write \(a = BC\), \(b = CA\), \(c = AB\), and let \(S\) be the centre and \(r\) the radius of its inscribed circle. Consider the line joining the midpoints of the sides \(BC\) and \(CA\). The internal bisector of the angle at \(A\) meets this line at a point \(D\), and the internal bisector of the angle at \(B\) meets it at a point \(E\). Determine the area of triangle \(SDE\).

The midpoints \(A_1\) of \(BC\) and \(B_1\) of \(CA\); the line \(A_1B_1\) carries \(D\) and \(E\), which need not lie on the segment \(A_1B_1\) itself.