(a) a fixed point on diagonal BD
In a rhombus, diagonals bisect each other. Suppose in the given rhombus ABCD, diagonals bisect each other at point O. Then, the distances of O from the four vertices A, B, C and D are equal. Thus, even if we take fixed point on diagonal BD as O1, O2, O3, O4, ..., they all are equidistant from the vertices A and C (by property of congruent triangles). Hence the locus of a point in rhombus ABCD which is equidistant from A and C is a fixed point on diagonal BD.
Alternatively, since diagonals of a rhombus bisect each at rt. ∠s, therefore, BD is the single bisector of AC, and therefore, all points equidistant from A and C lie on it.
