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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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Given In `DeltaABC` , P, Q and R are the mid-points of the sides BC, CA and AB respectively Also, `AD bot BC`.
To prove P, Q, R and D are concyclic.
Construction Join DR, RQ and QP
Proof In right angled `DeltaADP` , R is the mid-point of AB.
`:. RB=RD`
`rArr angle2=angle1` ....(i)
[angles opposite to the equal sides are equal]
Since, R and Q are the mid-point of AB and AC, then
`RQ||BC` [by mid-point theorem]
or `RQ||BP`
Since,` QP||RB`then quadrilateral BPQR is a parallelogram.
`rArr angle1=angle3` ..(ii)
[opposite angles of parallelogram are equal]
from Eqs. (i) and (ii),` angle2=angle3`
But `angle2+angle4=180^(@)` [linear pair axiom]
`:. angle3+angle4=180^(@) [:'angle2=angle3]`
Hence, quadrilateral PQRD is a cyclic quadrilateral.
So, points P, Q, R and D are con-cyclic.
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