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Mathematics

Properties of Cyclic Quadrilateral

ABCD is a cyc...

Question

ABCD is a cyclic quadrilateral, lines AB and DC intersect in the point F and lines Ad and BC intersects in the point E. Show that the circumference of $△ BCF and △ CDE$ intersect in a point G on the line EF.

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Solution

$∠ FBC = 90 ° (Angle in the semi circle is 90 °) Similarly, ∠ FGC = 90 ° ∴ ∠ FBC + ∠ FGC = 90 ° + 90 ° = 180 ° Hence, FBCG is a cyclic quadrilteral . Similarly, DEGC is a cyclic quadrilteral . ∠ FGC + ∠ EGC = 180 ° - ∠ FBE + 180 ° - ∠ EDF (FBGC and DEGC are cyclic quadrilateral) = ∠ ABE + ∠ ADF (Linear pair) = 180 ° (ABCD is a cyclic quadrilateral) Hence, FGE is straight line .$

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ABCD is a cyclic quadrilateral, lines AB and DC intersect in the point F and lines Ad and BC intersects in the point E. Show that the circumference of △BCF and △CDE intersect in a point G on the line EF.

ABCD is a cyclic quadrilateral, lines AB and DC intersect in the point F and lines Ad and BC intersects in the point E. Show that the circumference of $△ BCF and △ CDE$ intersect in a point G on the line EF.