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Properties of Cyclic Quadrilateral

54. Prove tha...

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54. Prove that the angle bisectors of the angles formed by producing opposite sides of cyclic quadrilateral (provided they are not parallel) intersect at right angles?

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Q. A cyclic quadrilateral ABCD whose opposite sides when produced intersect at the [points P and Q]. The bisectors PF and QF of

∠ P

∠ Q

respectively at F
produce QF to meet AB at E. Prove that

∠ P F G = 90^{\circ}

Q. The diagonals of a cyclic quadrilateral are at right angles. Prove that the perpendicular from the point of their intersection on any side when produced backwards, bisects the opposite side.

The quadrilateral formed by angle bisectors of a cyclic quadrilateral is also cyclic. Prove it.

Q. If bisectors of opposite angles of a cyclic quadrilateral

A B C D

intersect the circle, circumscribing it at the points

P

Q

P Q

is a diameter of the circle.

Q. If all the sides of a quadrilateral are produced in the same order (clockwise or anticlockwise).
Show that quadrilateral formed by the bisectors of exterior angles is cyclic.

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