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In a triangle ABC, if angle A =60 and altitudes from B and C meet AC and AB at P and Q respectively and intersect each other at I. Prove that APIQ and PQBC

are cyclic quadrilaterals.



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In quadrilateral AQIP, 
∠A=60 degree
∠AQI=∠API=90 degree
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 
∠1=30 degree
and in triangle AQC, angle BQC is a external angle so
∠2=30 degree
SO ∠1=∠2
So we can see that it is only possible when PQBC are cyclic qudrilateral.
4 4 4
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