Question

In a rhombus ABCD, M is any point in its interior as shown in the adjoining figure, such that MA = MC. Prove that B, M, D are collinear.

Answer 1

Given:

• ABCD is a rhombus.
• M is a point in its interior with MA = MC.

To Prove:

• B, M, D are collinear.

Proof [Step-wise]:

1. Let O be the intersection point of the diagonals AC and BD. In a rhombus, the diagonals bisect each other and are perpendicular, so OA = OC and AC ⟂ BD. Hence, BD is the perpendicular bisector of AC.
2. MA = MC means M is equidistant from A and C. The locus of points equidistant from A and C is the perpendicular bisector of segment AC.
3. From (1) BD is the perpendicular bisector of AC, and from (2) M lies on the perpendicular bisector of AC. Therefore, M lies on BD.
4. Hence, B, M, D are collinear.

B, M, D are collinear, as required.