Question

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

Answer 1

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

• Given,
• ∠ FBC = 90°  (angle in a semi-circle is 90°)
• Similarly, ∠ FGC = 90°
• ∴ ∠ FBC + ∠ FGC = 90° + 90° = 180°
• Therefore, FBCG is a cyclic quadrilateral
• Similarly,
• ∠ EDC = 90°  (angle in a semi-circle is 90°)
• Similarly, ∠ EGC = 90°
• ∴ ∠ EDC + ∠ EGC = 90° + 90° = 180°
• Therefore, DEGC is a cyclic quadrilateral
• From above equations, we have,
• ∠ FGC + ∠ EGC = 180° - ∠ FBE + 180° - ∠ EDF
• = ∠ ABE + ∠ ADF (linear pair)
• = 180°  (as ABCD is a cyclic quadrilateral)
• Hence, FGE is a straight line