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Angle Bisectors On Circumcircle: What is this about?
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Copyright © 1996-2008 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.
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By a the theorem of the secant angles (or with the help of the Exterior Angle Theorem),
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FIM = ACI + CAI = C/2 + A/2.
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It follows that in ΔFIM, angles at F and I add up to 90°:
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A/2 + B/2 + C/2 = 180°/2 = 90°.
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We conclude that the remaining angle at M is necessarily right.
References
- T. Tao, Solving Mathematical Problems, Oxford University Press
Copyright © 1996-2008 Alexander Bogomolny
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