- Dear Angel

Beautiful..... !!!!

Thanks

Antreas

http://amontes.webs.ull.es/geogebra/Hyacinthos21421.html

>

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

> http://amontes.webs.ull.es/geogebra/Hyacinthos21421Pedal.html

>

>

>

>

> http://amontes.webs.ull.es/geogebra/Hyacinthos21428.html

>

- It is Another "Seven Circles Theorem" !

The "Seven Circles Theorem" states:

Let a,b',c, a',b,c' be a closed chain of six circles,

all toouching a base circle w[omega], and suppose that

their points of contact with w are six distinct points

A,B',C,A',B,C' respectively. Then, subject of a certain

extra condition to be discussed below, AA',BB',CC' are concurrent.

(C J A Evelyn, B G Money - Coutts, J A Tyrrell:

The Seven Circles Theorem and other theorems. London 1974, p. 31)

See also

http://en.wikipedia.org/wiki/Seven_circles_theorem

http://mathworld.wolfram.com/SevenCirclesTheorem.html

APH

--- In Hyacinthos@yahoogroups.com, "Angel" <amontes1949@...> wrote:

>

> Dear Antreas

>

> --- In Hyacinthos@yahoogroups.com, "Antreas" wrote:

> >

> > Dear Alex

> >

> > Angel has tested the loci problems and it seems that

> > for all P's the lines are concurrent for the cevian case,

> > and for the pedal case as well except for the points P on the

> > bisectors and on the circumcircle.

> >

> > Hmmmmmmm...... Maybe it is true in general ie for any

> > circle intersecting the sidelines of the triangle!!

> >

>

>

> A check with GeoGebra:

>

>

>

> Let ABC be a triangle and P a point.

>

> Let (Q) be the CEVIAN circle of P and (Qab), (Qac) the circles touching AB,AC and (Q) internally and let (Tab), (Tac) be the points of contact. Similarly we define the points Tbc, Tba and Tca, Tcb.

>

> For all P (except for the points P on the bisectors) the lines TabTac, TbcTba, TcaTcb are concurrent.

>

> http://amontes.webs.ull.es/geogebra/Hyacinthos21421.html

>

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

>

> Let (Q) be the PEDAL circle of P and (Qab), (Qac) the circles touching AB,AC and (Q) internally and let (Tab), (Tac) be the points of contact. Similarly we define the points Tbc, Tba and Tca, Tcb.

>

> For all P (except for the points P on the

> bisectors and on the circumcircle) the lines TabTac, TbcTba, TcaTcb are concurrent.

>

> http://amontes.webs.ull.es/geogebra/Hyacinthos21421Pedal.html

>

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

>

> Let (Q) be the circle Intersecting the sidelines of the triangle in D, E and F. (Qab), (Qac) the circles touching AB,AC and (Q) internally and let (Tab), (Tac) be the points of contact. Similarly we define the

> points Tbc, Tba and Tca, Tcb.

>

> The lines TabTac, TbcTba, TcaTcb are concurrent.

>

> http://amontes.webs.ull.es/geogebra/Hyacinthos21428.html

>

> Best regards,

> Angel M.

>