Question

P,Q and R are the mid points of sides BC,CA, and AB of ABC, and AD. is . the perpendicular from A to BC. prove that P,Q,R and D are concyclic.

Answer 1

We have to prove that $R,D,P$ and $Q$ are concylic.

Refer image,

Join $RD,QD,PR$ and $PQ$
$\because RP$ joints $R$ and $P$, the mid-point of $AB$ and $BC$
$\therefore RP||AC$ [Mid-point theorem]

Similarly, $PQ||AB$

$\therefore ARPQ$ is a $||gm$

So $\angle RAQ=\angle RPQ$ [Opposite angles of a $||gm$]

$\because ABD$ is a rt. $\angle D\Delta$ and $DR$ is a median,

$\therefore RA=DR$ and $\angle 1=\angle 2$........(2)

Similarly $\angle 3=\angle 4$........(3)

Adding (2) and (3), we get,

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

$\angle RDQ=\angle RAQ$

$\angle RPQ$ [Proved above]

Here, $R,D,P$ and $Q$ are conyclic

[$\because \angle D$ abd $\angle P$ are substended by $RQ$ on the same soide of it]