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## Concurrency of four segments joining four points and their isogonal conjugates.

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• Let us call, for reasons discussed shortly (*), a simple quadrilateral that has its opposite sides (or their extensions) going through two of the vertices of a
Message 1 of 2 , May 2, 2007
Let us call, for reasons discussed shortly (*), a simple
quadrilateral that has its opposite sides (or their extensions) going
through two of the vertices of a triangle ABC and its opposite
vertices on isogonal lines through the third vertex, a "perspective
parallelogram".

The four segments, joining the four vertices of a "perspective
parallelogram" with their four isogonal conjugates, are concurrent.

(*) If we perceive the three vertices of a triangle as the three
vanishing points of a three point perspective drawing (i.e., perceive
the three sides of the triangle as the three horizon lines of that
drawing), the statement of the message becomes self-evident. For a
brief discussion on the application of this view on similar problems,
see message 15247.
• We now consider the general case, of any pair of “perspective parallelograms”, i.e. of any pair of simple quadrilaterals that have their opposite sides (or
Message 2 of 2 , May 2, 2007
We now consider the general case, of any pair of âperspective
parallelogramsâ, i.e. of any pair of simple quadrilaterals that have
their opposite sides (or their extensions) going through the same two
vertices of a triangle ABC and their opposite vertices on the same
isogonal lines through the third vertex.

The segments, joining the opposite vertices of any pair of
âperspective parallelogramsâ, are concurrent.

As in the special case, if we perceive the three vertices of a
triangle as the three vanishing points of a three point perspective
drawing (i.e., perceive the three sides of the triangle as the three
horizon lines of that drawing), the statement of the message becomes
self-evident.
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