## 17327Re: NPC and Kiepert hyperbola

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• Mar 1 10:28 AM
Dear Francisco.

I understand what you are saying.
It is an interesting question.
Everything is allright for me. Except for your statement "The line
PQR here is the parallel to DP', LP", etc. through P." That is just
the point you started with. Under what conditions is there a line PQR?
For a better understanding let us name:
 Line L = the line through Q and R
 Pi = Infinity Point, being the IntersectionPoint of parallel
lines DP', LP'', etc.
We started with the question under what conditions P,Q,R are
collinear?
You noticed that when P is a point on the Kiepert Hyperbola that in
that case P,Q,R are collinear.
This is easy to be confirmed with drawings.
Now you state that when P is on NPC, then Q and R coincide with
Infintypoint Pi.
Then you state (in other words): so P.Q.R = P.Pi.Pi = P.Pi, ergo they
are collinear.
Is that right?
However the question is not to draw the line P.Pi. The original
question is to draw a line through Pi.Pi and then look if this line
coincides with P.
When you draw a line through Pi.Pi you actually draw a line through
Pi (evident) but you don't know if it comes from the right or the
left side of the plane.
It can be any line parallel to lines DP', LP'', etc. There is an
endless amount of these lines possible.
As soon as we found that very special line that is inherent to the
problem then we have to review if P is on this line.
Of course this should be confirmed with drawings (as in real life we
should try to confirm all things that are said to be true).
When I make drawings in Cabri and take a variable point P and move it
to any point on NPC I see what happens to QR in these drawings:
1. It is moving to a parallel line to DP', LP'', etc.
2. It is going through the point T on the BrocardAxis that I
described in earlier mail.
3. It is NOT going through P !
So this is a confirmation of the contrary of your statement. For me
it is clear that P,Q,R are not collinear when P is on NPC, except for
the situation when it is an intersectionpoint of NPC with the Kiepert
Hyperbola.
Best regards,
Chris van Tienhoven

p.s. There is a comparable situation when you take 2 points Q and R
on a circle and any point P outside the circle and you want to know
when P,Q,R are collinear. As soon as Q and R coincide you notice that
in this case (inherent to the problem) QR is the tangent to the
circle and P is not on the tangent.
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