The Mirror Property of Altitudes via Pascal's Hexagram: What is this about?
A Mathematical Droodle

Explanation

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, 2007, submitted for publication

Pascal and Brianchon Theorems

• Pascal's Theorem
• Pascal in Ellipse
• Pascal's Theorem, Homogeneous Coordinates
• Projective Proof of Pascal's Theorem
• Pascal Lines: Steiner and Kirkman Theorems
• Brianchon's theorem
• Brianchon in Ellipse
• The Mirror Property of Altitudes via Pascal's Hexagram
• Pappus' Theorem
• Pencils of Cubics
• Three Tangents, Three Chords in Ellipse
• MacLaurin's Construction of Conics
• Pascal in a Cyclic Quadrilateral
• Parallel Chords
• Parallel Chords in Ellipse
• Construction of Conics from Pascal's Theorem
• Pascal: Necessary and Sufficient
• Diameters and Chords
• Chasing Angles in Pascal's Hexagon