Question

Question 3
If P, Q and R are the mid-points of the sides BC, CA and AB of a triangle and AD is the perpendicular from A on BC, prove that P, Q, R and D are concyclic.

Answer 1

R is mid point of hypotenuse AB of right $\Delta ADB$.
$\therefore RB=RD$ [midpoint of hypotenuse is equidistant from the three vertices of $\Delta$]
$\Rightarrow \angle 2=\angle 1$ ...(I) (isosceles $\Delta$ property)
RQ joins midpoints of sides AB and AC.
$\therefore$ RQ || BC (midpoint theorem)
$\Rightarrow$ RQ || BP
Similarly, QP || RB
$\therefore BPQR$ is a parallelogram.
$\angle 1=\angle 4$ ...(II) (opposite angles of a parallelogram)
From (I) and (II),
$\angle 2=\angle 4$
but, $\angle 2+\angle 3={180}^{\circ }$ (linear pair axiom)
$\therefore \angle 3+\angle 4={180}^{\circ }$
$\Rightarrow PQRD$ is cyclic because sum of the opposite angles is ${180}^{\circ }$ .
Therefore, points P, Q, R, D are concyclic.