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• ## Re: [EMHL] Property of GBC, GCA, GAB

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• You are right. I don t see the link between your question and the answer of M.Rousee except both are about affine geometry. In fact with your notations the
Message 1 of 9 , Jun 30, 2005
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You are right.
except both are about affine geometry.

In fact with your notations the point Pi divides the segment OiHi in
the same ratio for i=0,1,2,3 and that is affine geometry! (For i=0, I
mean Po =P, Oo=O and Ho=H).
For example for k=2 with your notations, Pi=Gi, the center of gravity
or isobarycenter (think about Euler lines!).
So it suffices to prove your theorem for 2 special values of k;
For k=0, Pi=Oi the circumcenter and you must prove that O is the
center of gravity or the isobarycenter of O1O2O3.
For k=1, Pi=Hi the orthocenter and you must prove that H is the center
of gravity or the isobarycenter of H1H2H3.
You can also choose k=-2 where Pi=Gi the center of gravity.
I am surprised you don't know the notion of barycenters which is the
core of affine geometry.
Friendly
François
• I correct a little typo in my last answer. With your notation: OP:PH = k:1-k you mean: OP = k OH so P = G for k=1:3 (and not for k=-2!) Friendly François
Message 1 of 9 , Jul 1, 2005
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I correct a little typo in my last answer.
OP:PH = k:1-k
you mean:
OP = k OH
so P = G for k=1:3 (and not for k=-2!)
Friendly
François
• ... Yes, it s an error of mine. I have forgotten to change the subject. Etienne [Non-text portions of this message have been removed]
Message 1 of 9 , Jul 1, 2005
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>You are right.
>I don't see the link between your
>question and the answer of M.Rousee
>except both are about affine geometry.

Yes, it's an error of mine.
I have forgotten to change the subject.

Etienne

[Non-text portions of this message have been removed]
• Dear friends Beeing given 3 lines L_{1}, L_{2}, L_{3} in R^3 with equations: x = a_{k} z + b_{k} y = c_{k} z + d_{k} for k= 1, 2, 3 1° Is there a simple way
Message 1 of 9 , Jul 6, 2005
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Dear friends
Beeing given 3 lines L_{1}, L_{2}, L_{3} in R^3 with equations:
x = a_{k} z + b_{k}
y = c_{k} z + d_{k}
for k= 1, 2, 3
1° Is there a simple way to write down an equation of the ruled
quadric having these 3 lines as generators?
2° Is it possible to write easily equations for the two systems
Friendly
François
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