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In Delta ABC, P,Q and R are midpoints o...

In `Delta ABC, ` P,Q and R are midpoints of sides AB,AC and BC respectively. Seg `AS _|_BC` and PQ || BR side BC. Prove that `square ` PQRS is a cyclic quadrilateral.

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P and Q are the midpoints of sides AB and AC of `Delta ABC.`
`:. ` by the midpoit theorem.
`PQ || BC` ( i.e. BR ) and PQ `= (1)/(2) BC ` …..(1)
`BR = (1)/(2) BC ` ….( R is the midpoint of side BC ) …(2)
From (1) and (2),
`PQ ||BR ` and `PQ =BR`.
`:. square ` PQRB is a parallelogram
`:. /_ B = /_ Q ` ...(Opposite angles of a parallelogram ) ...(3)
In right angled triangle, median to the hypotenuse is half of the hypotenuse.
`:. ` in `Delta ABS, SP = (1)/(2) AB`.
`PB = (1)/(2) AB ` ...( P is the midpoint of side AB )
`:. SP = PB `
`:. /_ B = /_ PSB ` ...( Isosceles triangle theorem ) ...(4)
From (3) and (4),
`/_ Q = /_ PSB ` ....(5)
`/_ PSB + /_ PSR= 180^(@)` ....( Angles in a linear pair ) ...(6)
From (5) and (6)
`/_ Q + /_ PSR = 180^(@)`
`:. ` by the converse of cyclic quadrilateral theorem , `square ` PQRS is a cyclic quadrilateral.
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