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P ,\ Q\ a n d\ R are, respectively, the ...

`P ,\ Q\ a n d\ R` are, respectively, the mid points of sides `B C ,\ C A\ a n d\ A B` of a triangle `A B C`

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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.)
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