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Projections on Internal and External Angle Bisectors: What Is This About?
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Copyright © 1996-2008 Alexander Bogomolny
The applet below provides an illustration to a problem from an outstanding collection by T. Andreescu and R. Gelca:
| Prove that the four projections of vertex A of ΔABC on the exterior and interior angles bisectors at B and C are collinear. |
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Let M and N denote projections of A onto the interior angle bisectors at B and C, and let P and Q be the projections on the respective exterior angle bisectors. Let us prove that P lies on MN. Since the interior and exterior angles bisectors at a vertex of a triangle are perpendicular, quadrilateral AMBP is a rectangle. Hence
 AMP = ABP.
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We have
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If I is the incenter, quadrilateral ANIM is cyclic for it has two opposite right angles. Therefore,
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AMN = AIN.
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But AIN is exterior to Δ; hence AIN = A/2 + B/2 = AMN. Which implies that points M, N, and P are collinear. That Q lies on the same line is shown in a similar manner.
(Nathan Bowler has observed that P, N and M are collinear since they lie on the Simson line of A with respect to BIIC.)
References
- T. Andreescu, R. Gelca, Mathematical Olympiad Challenges, Birkhäuser, 2004, 5th printing, 1.2.4 (p. 8)
Copyright © 1996-2008 Alexander Bogomolny
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