The Mirror Property of Altitudes via Pascal's Hexagram
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A Mathematical Droodle


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Explanation

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The applet may suggest the following statement by Greg Markowsky:

Let points C and D be located on a semicircle with diameter AB. Let E be the intersection of AD and BC and F the foot of the perpendicular from E to AB. Then EF is the bisector of angle CFD.


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Note that since inscribed angles ACB and ADB subtend a diameter they are right. So that AD and BC are altitudes in ΔABS where S is the intersection of AC and BD, E is the orthocenter of this triangle and EF is the third altitude. What is claimed is nothing more nor less than the mirror property of the altitude. Greg's shows how the latter can be derived from Pascal's theorem of the hexagram.

Proof

Reflect C and D in AB to obtain C' and B'. Consider the hexagon ADC'BCD'. By Pascal's theorem, the three points at which the pairs {AD, BC}, {DC', CD'}, and {C'B, D'A} of opposite sides meet, lie on a straight line. The three intersection points are E, F, and E'. This is clear for E and E'. Assuming that DC' and CD' intersect in F', observe that, by symmetry, EE' is orthogonal to AB and, for the same reason, DC' and CD' intersect on AB, the axis of symmetry. Therefore both F and F' lie on AB and also on EE' and, thus, coincide.

References

  1. G. Markowsky, Pascal's Hexagon Theorem implies the Butterfly Theorem, Mathematics Magazine, Volume 84, Number 1, February 2011 , pp. 56-62(7)

Pascal and Brianchon Theorems

Related material
Read more...

  • The Mirror Property of Altitudes
  • The Angle Bisectors
  • Optical Property of Ellipse
  • Fagnano's Problem
  • Parabolic Mirror, Illustration
  • Existence of the Orthocenter
  • All about altitudes
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