Question

quadrilateral ABCD is cyclic quadrilateral, lines AB and DC inart
in the point F and lines AD and BC intersect in the point
E. Show that the circumcircles of ABCF and ACDE
intersect in a point G on the line EF.​

Answer 1

Quadrilateral = ABCD (Given)

Lines of the Quadrilateral = AB and DC (Given)

∠FBC=90°  ( As angle in the semi circle is 90°)

Similarly,

∠FGC=90°  

Therefore,

∠FBC+∠FGC = 90°+90° = 180°

Hence, FBCG  is a cyclic quadrilteral.

Similarly, DEGC is a cyclic quadrilteral.

∠FGC+∠EGC = 180°−∠FBE + 180°−∠EDF    

Since, FBGC and DEGC are cyclic quadrilateral, thus

∠ABE + ∠ADF =180° ( Linear Pair)  

Hence, FGE is straight line.