Question

In the figure below, two vertices of a parallelogram are joined to the mid points of two sides.

Show that the lines divide the diagonal given into 3 equal parts.

Answer 1

In parallelogram ABCD, M is the midpoint of AB, N is the midpoint of DC.

$\therefore$ AM = MB, DN = NC

In parallelogram ABCD, AB = CD

$\frac{AB}{2}=\frac{CD}{2}=AM=DN$

$\therefore$ AM = BM = DN = NC .....(i)

DM and BN are parallel lines.

Draw 2 lines through A & C parallel to DM.

From (1), these parallel lines are equidistant

Thus, parallel lines DM & BN divide AC into 3 equal parts.

AP = PQ = QC