Question
Fri March 13, 2015 By: Pragya Chaturvedi

in a triangle ABC angle A is 60 and the altitudes from. B and C meet AC and AB at P and Q and intersect each other at I prove that APIQ and PQBC are cyclic quadrilaters

Expert Reply
Vimala Ramamurthy
Sat March 14, 2015
C o n s i d e r space t h e space f o l o w i n g space f i g u r e.
C o n s i d e r space t h e space t r i a n g l e s space triangle A B P space a n d space triangle A C Q W E space k n o w space t h a t space a n g l e s space angle B P A space equals space angle C Q A equals 90 degree T h e r e f o r e comma angle A B P equals 90 degree minus angle B A P equals 90 degree minus 60 degree equals 30 degree a n d angle A C Q equals 90 degree minus angle C A Q equals 90 degree minus 60 degree equals 30 degree  C o n s i d e r space t h e space t r i a n g l e s space triangle B I Q space a n d space triangle C I P W e space k n o w space t h a t space a n g l e s space angle B Q I space equals space angle C P I equals 90 degree T h e r e f o r e comma angle B I Q equals 90 degree minus angle I B Q equals 90 degree minus angle A B P equals 90 degree minus 30 degree equals 60 degree a n d angle C I P equals 90 degree minus angle I C P equals 90 degree minus angle A C Q equals 90 degree minus 30 degree equals 60 degree  S i n c e space v e r t i c a l l y space o p p o s i t e space a n g l e s space a r e space e q u a l comma space angle Q I P equals angle B I C... left parenthesis 1 right parenthesis T h e space a n g l e space a r o u n d space a space p o i n t space i s space 360 degree T h u s space t h e space a n g l e space a r o u n d space I equals 360 degree rightwards double arrow angle B I Q plus angle B I C plus angle C I P plus space angle P I Q equals 360 degree rightwards double arrow 60 degree plus angle B I C plus 60 degree plus space angle P I Q equals 360 degree rightwards double arrow 120 degree plus 2 angle B I C equals 360 degree space space space space space space space space space space space left square bracket f r o m space e q u a t i o n space left parenthesis 1 right parenthesis right square bracket rightwards double arrow 2 angle B I C equals 360 degree minus 120 degree rightwards double arrow 2 angle B I C equals 240 degree rightwards double arrow angle B I C equals fraction numerator 240 degree over denominator 2 end fraction rightwards double arrow angle B I C equals 120 degree rightwards double arrow space angle Q I P equals angle B I C equals 120 degree space space space space space space..... left parenthesis 2 right parenthesis  N o w space c n s i d e r space t h e space q u a d r i l a t e r a l space A Q I P. angle A equals 60 degree space a n d space angle Q I P equals 120 degree space rightwards double arrow angle A plus space angle Q I P equals 60 degree plus 120 degree equals 180 degree A l s o comma space angle A Q I plus angle A P I equals 90 degree plus 90 degree equals 180 degree T h e r e f o r e comma space t h e space q u a d r i l a t e r a l space A Q I P space i s space a space c y c l i c space q u a d r i l a t e r a l.
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