Angle Bisectors On Circumcircle

What is this about?
A Mathematical Droodle

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Explanation

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Copyright © 1996-2012 Alexander Bogomolny

The applet attempts to suggest the following problem [Tao, pp. 50-51]:

ABC is a triangle that is inscribed in a circle. The angle bisectors of A, B, C meet the circle in D, E, F, respectively. Show that AD is perpendicular to EF.

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We'll concentrate on ΔFIM. By a theorem of the inscribed angles,

∠IFM = ∠CFE = ∠CBE = ∠B/2.

By a the theorem of the secant angles (or with the help of the Exterior Angle Theorem),

∠FIM = ∠ACI + ∠CAI = ∠C/2 + ∠A/2.

It follows that in ΔFIM, angles at F and I add up to 90°:

∠A/2 + ∠B/2 + ∠C/2 = 180°/2 = 90°.

We conclude that the remaining angle at M is necessarily right.

References

1. T. Tao, Solving Mathematical Problems, Oxford University Press

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Copyright © 1996-2012 Alexander Bogomolny