1274. Angle bisectors...

    Dear Teacher:

    ¿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...

    Dear Teacher:

    Since

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...