viernes, 24 de febrero de 2017

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...


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