Dear friends

I have noticed that the Nikos configuration in message 6466 was simply the

figure of the special Sondat theorem that is to say 2 orthologic triangles

ABC and A'B'C' with the same orthology center O.

Then :

1°ABC and A'B'C' are perspective wrt some perspector S

2°Line SO is orthogonal to the perspectrix.

Nikos only prove the 1st point and I have said I don't like his proof very

much.

In the next message n°6467, Darij noticed that ABC and A'B'C' are conjugate

wrt some circle of center O and that implies the 1st point (and also the 2st

as we shall see).

But Darij regretted this proof was not valid if the circle is not real.

I don't agree with him for polarity is in fact defined wrt a quadratic form

before beeing defined wrt the conic possibly vanishing the form.

So Darij proof is valid in all cases.

Now if we come back to the general situation of 2 triangles ABC and A'B'C'

conjugate wrt some conic {Gamma}, then it is well known that ABC and A'B'C'

are perspective with perspector S and perspectrix L. Moreover L is the polar

line of S wrt conic {Gamma}.

In the special Sondat configuration, as {Gamma} is a circle of center O and

L is the polar line of S wrt circle {Gamma}, then line SO is orthogonal to L

and we are done with the 2st point.

So I think Darij proof is The Synthetic Solution of the special Sondat

Theorem.

Friendly

François

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