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Locus connected with the Simson point
Posted:
Jul 31, 2009 8:11 AM


It all started with an interesting theorem from
Paul Yiu, Heptagonal triangles and their companions, Forum Geometricorum Volume 9 (2009), 125148
which stated that: In a regular heptagon ABCDEFG the Simson lines for B, C and G in respect to the triangle AEF are concurrent.
This got me thinking about the locus of all points P whose Simson likes with respect to a given triangle ABC pass through a given point S. A straightforward calculation seems to suggest that said locus is a hyperbole, if I did not mess up the calculations.
Does someone have a reference for this?
A more interesting question arises with respect to the abovementioned Yiu's theorem. If indeed the locus is a hyperbole then B, C and G lie on it. Hence there should be a fourth point on the circle on which B,C and G lie whose Simson line also passes through the same point.
Which is this point?



