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.