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, then prove that P, Q, R and D are concyclic.

Explanation

We have to prove that R,D,P and Q are concyclic.
Join RD, QD,PR and PQ.
Since, RP joins R and P, the mid-point of AB and BC.
`therefore RP|\ |AC` (mid-point theorem)
Similarly, `PQ|\ |AC`
Therefore, ARPQ is parallelogram.
So, `angle RAQ=angle RPQ` (opposite angles of a parallelogram.....(1)
Since, ABD is a right-angled triangle and DR is a median.
`therefore RA=DR` and `angle1 =angle2` ......(2)
Similarly , `angle3=angle4`........(3)
Adding eqs. (2) and (3) , we get
`angle1 +angle3 =angle2 +angle4`
`rArrangle RDQ=angle RAQ`
`=angleRPQ` (proved above)
Hence R,D,P and Q are concyclic.
(`becauseangleD` and `angleP` are subtended by RQ on the same side of it.)