- Dear Sung Hyun,

Do you mean if we have three arbitrary circles

(A,a), (B,b), (C,c) and the points A1, B1, C1

on them rotate with constant anglular velocity

then the locus of the centroid of A1B1C1

is a circle with center the centroid of ABC?

Best regards

Nikos Dergiades

> Given three circles with three points

> moving on the circles with equal

> angular velocity, when is the locus of the circumcenter a

> circle?

>

> I have verified that it indeed is a circle for some cases.

>

> Was such generalization posed before?

>

> Sung Hyun

>

>

> [Non-text portions of this message have been removed]

>

>

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> - Dear Sung Hyun and Nikolaos
> Do you mean if we have three arbitrary circles

This is nothing else than the linearity of a vectorial rotation (the common angular velocity doesn't need to be constant)

> (A,a), (B,b), (C,c) and the points A1, B1, C1

> on them rotate with constant anglular velocity

> then the locus of the centroid of A1B1C1

> is a circle with center the centroid of ABC?

>

> Best regards

> Nikos Dergiades

Suppose that at t=0, A1,B1,C1 are A0,B0,C0.

Then the vectorial rotation mapping AA0,BB0,CC0 respectively to AA1,BB1,CC1 will map GG0 to GG1 which means that G1 moves on the circle (G,GG0) wit the common angular velocity.

It think that the circumcenter of A1B1C1 moves on a circular quadric.

Friendly. Jean-Pierre

>

> > Given three circles with three points

> > moving on the circles with equal

> > angular velocity, when is the locus of the circumcenter a

> > circle?

> >

> > I have verified that it indeed is a circle for some cases.

> >

> > Was such generalization posed before?

> >

> > Sung Hyun

> >

> >

> > [Non-text portions of this message have been removed]

> >

> >

> >

> > ------------------------------------

> >

> > Yahoo! Groups Links

> >

> >

> > Hyacinthos-fullfeatured@yahoogroups.com

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> - Dear Jean-Pierre and Nikolaos,

I never mentioned the locus of centroid, but I mentioned the locus of

centroid.

As Jean-Pierre said, the locus of circumcenter is not a circle in general,

but I am saying that it seems to approach circle (very closely) for certain

configuration of initial points (where they start rotating from) and I was

asking, for what condition would the locus be a circle?

Regards,

Sung Hyun

[Non-text portions of this message have been removed] > I never mentioned the locus of centroid, but I mentioned the locus of

This is embarassing. I meant that I mentioned the locus of circumcenter not centroid. How stupid...

> centroid.

Sorry for the confusion.

Sung Hyun

--- In Hyacinthos@yahoogroups.com, Lim Sung Hyun <bbbbbblow@...> wrote:

>

> Dear Jean-Pierre and Nikolaos,

>

> I never mentioned the locus of centroid, but I mentioned the locus of

> centroid.

>

> As Jean-Pierre said, the locus of circumcenter is not a circle in general,

> but I am saying that it seems to approach circle (very closely) for certain

> configuration of initial points (where they start rotating from) and I was

> asking, for what condition would the locus be a circle?

>

> Regards,

> Sung Hyun

>

>

> [Non-text portions of this message have been removed]

>