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• Dear Darij, ... My first question in previous message was about adjusting original ABCD to make it homothetic to Alexey s A B C D . This adjustment will
Message 1 of 41 , Jan 15, 2004
Dear Darij,

>
> Alas, I don't understand 1. and 2., and have no
> answers to 3. and 4.; however, the analogy with
> centroids would be quite unusual. I am still
> hoping to find a nice expression for the radius
> of (O'), but I don't see any direct way to this.
>

original ABCD to make it homothetic to Alexey's A"B"C"D".
This 'adjustment' will move the center of the original incircle
onto line O'P. Will it change it's radius also?
Now, along with incenters X,Y,Z,W we also looked at the
ex-centers Xe,Ye,Ze,We. Quadrilateral XeYeZeWe is homothetic to
XYZW (?). When we 'adjust' side AB to A1B1, A1B1 will be tangent
still to circle X and the new incircle of A1B1C1D1.
The new side A1B1 will not be tangent to ex-circle as old AB was.
The configuration of ex-circles has to 'adjust' all the way up.
But if XeYeZeWe was homothetic to XYZW before, it will not
be after this 'ripple'. Will it?

The second one was about the possible ex-circle that touches
the continuations of all 4 sides of ABCD . The 'adjusted' A1B1C1D1
being bicentric, maybe has a better chance of having
this ex-circle.

Sincerely,
Rafi.
• Dear Darij, ... My first question in previous message was about adjusting original ABCD to make it homothetic to Alexey s A B C D . This adjustment will
Message 41 of 41 , Jan 15, 2004
Dear Darij,

>
> Alas, I don't understand 1. and 2., and have no
> answers to 3. and 4.; however, the analogy with
> centroids would be quite unusual. I am still
> hoping to find a nice expression for the radius
> of (O'), but I don't see any direct way to this.
>

original ABCD to make it homothetic to Alexey's A"B"C"D".
This 'adjustment' will move the center of the original incircle
onto line O'P. Will it change it's radius also?
Now, along with incenters X,Y,Z,W we also looked at the
ex-centers Xe,Ye,Ze,We. Quadrilateral XeYeZeWe is homothetic to
XYZW (?). When we 'adjust' side AB to A1B1, A1B1 will be tangent
still to circle X and the new incircle of A1B1C1D1.
The new side A1B1 will not be tangent to ex-circle as old AB was.
The configuration of ex-circles has to 'adjust' all the way up.
But if XeYeZeWe was homothetic to XYZW before, it will not
be after this 'ripple'. Will it?

The second one was about the possible ex-circle that touches
the continuations of all 4 sides of ABCD . The 'adjusted' A1B1C1D1
being bicentric, maybe has a better chance of having
this ex-circle.

Sincerely,
Rafi.
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