`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` and AD is the perpendicular from vertex A to BC,then prove that the points P,Q,R and D are cyclic.

Explanation

In `/_\ABC P,Q \and\ R` are the midpoints of the sides BC,CA and AB respectively.
Also,`AD⊥BC`In a right-angled triangle, `ADP,R` is the midpoint of AB
`RB=RD`
`/_2=/_1`(1)
since angles opposite to the equal sides are equal.
Since,`R and Q`are the mid-points of AB and AC,
then `RQ∥BC \or \ RQ∥BP` (by mid-point theorem)
Since,`QP∥RB` then quadrilateral BPQR is a parallelogram,
`=>/_1=/_3`(2)
since angles opposite to the equal sides are equal.
From equations (1) and (2),`/_2=/_3`
But`/_2+/_4=180^0`(by linear pair axiom)
`/_3+/_4=180^0(/_2=/_3)`
Hence ,quadrilateral `PQRD` is a cyclic quadrilateral.