¿Do the angle bisectors of a quadrilateral form a cyclic quadrilateral?
I answered Pepe Chapuzas that not always, because the angle bisectors of a square formed... a point! But if the angle bisectors did form a quadrilateral then this was really a cyclic quadrilateral!
Nina Guindilla rushed to do a proof...
2α + 2β + 2γ + 2δ = 360°then
α + β + γ + δ = 180°
The upper blue angle is
180° − β − γ
and the lower blue angle is
180° − α − δ
and the sum of both blue angles sall be
360° − α − β − γ − δ = 360° − 180° = 180°
That is, the quadrilateral formed by the angle bisectors is cyclic.
Furthermore, the angle bisectors of squares, rhombi, kites and darts do not form any quadrilateral (but a point). Here are some examples of cyclic quadrilaterals formed by angle bisectors:
A square from a rectangle.
A rectangle from a rhomboid.
A kite from an isosceles trapezoid.
An isosceles trapezoid from a...