I a triangle ABC if angle A =60 and the altitudes from Band C meet AC & AP at P and Q respectively and intersect each other at I prove that APIQ and PQBC and are cyclic quadrilaterals hence find the measure of angle BIC

In quadrilateral AQIP, A=60 degreeAQI=API=90 degreeSo PIQ=360-90-90-60=120 degreeSo we can see that in  quadrilateral AQIP ,A+PIQ=180 and AQI+API=180SO APIQis a cyclic quadrilateral.Now in triangle ABP, angle APC is a external angle So 90=60+11=30 degreeand in triangle AQC, angle BQC is a external angle so 90=60+22=30 degreeSO 1=2So we can see that it is only possible when PQBC are cyclic qudrilateral.

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