In quadrilateral AQIP,
So ∠PIQ=360−90−90−60=120 degree
So we can see that in quadrilateral AQIP ,
∠A+∠PIQ=180 and ∠AQI+∠API=180
SO APIQ is a cyclic quadrilateral.
Now in triangle ABP, angle APC is a external angle
and in triangle AQC, angle BQC is a external angle so
So we can see that it is only possible when PQBC are cyclic qudrilateral.