Question

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

Prove that P,Q,R and D are concyclic

It is given that 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.

Now Join RD, QD, PR and PQ. RP joins the mid-point of AB,R, and the mid-point of BC, that is P.

From the midpoint theorem

$RP\left|\right|AC$ Likewise $PQ\left|\right|AB$

Hence ARPQ is a parallelogram.

So,$\angle RAQ=\angle RPQ$ [Opposite angles of a parallelogram]…(1)

ABD is a right-angled triangle and DR is a median,

$RA=DR$ since $DR$ divides the base into equal parts.

and $\angle 1=\angle 2$…(2) since $\angle ARD=90°$ and the sum of the angles of a triangle is $180°$.

Similarly $\angle 3=\angle 4$ …(3)

On adding equations (2) and (3),

$\angle 1+\angle 3=\angle 2+\angle 4$

$\Rightarrow \angle RDQ=\angle RAQ$

Since $\angle D$ and $\angle P$ are subtended by RQ on the same side of it, we get the points R, D, P and Q concyclic.

Hence, R, D, P and Q are concyclic.