- We have recently finished writing up a several year project in

geometry titled "Synthetic Cevian Geometry". Starting from work of

Grinberg, Ehrmann, and Yiu in these Hyacinthos messages (Hyacinthos

#6423, #959, #1790) we have been able to find synthetic proofs of many

new results in the theory of cevian triangles. One important result

that we have obtained is a generalized Feuerbach theorem.

Let P be a point not on the sides of a triangle ABC or its

anti-complementary triangle, and let Q denote its isotomcomplement =

the complement of the isotomic P' of P. We define a generalized

orthocenter H depending on P, as follows. Let DEF be the cevian

triangle of P with respect to ABC. Then H is defined as the

intersection of the lines AH, BH, CH through the vertices A, B, C

which are parallel, respectively, to the lines QD, QE, QF. We have

proved that H lies on the conic ABCPQ. We have also proved the

following affine formula for the point H. Let the cevian triangle of

P' be D'E'F' and let the mapping T_2 be the unique affine map taking

triangle ABC to D'E'F'. If K denotes the complement mapping, then

H = K^-1 * (T_2)^-1 * K(Q).

The related point K(H) = O = (T_2)^-1 * K(Q) is called the generalized

circumcenter of P, and is the center of a circumconic C_O of ABC which

can be given as

C_O = (T_2)^-1(N_P'),

where N_P' is the 9-point conic of the quadrangle ABCP'. This formula

is proved by showing that C_O is the 9-point conic of the quadrangle

A_2B_2C_2Q, where A_2B_2C_2 =

(T_2)^-1(ABC) is the anticevian triangle of Q with respect to ABC. We

also show that the complement of the conic C_O is the 9-point conic

N_H of the quadrangle ABCH. Let I_P be the inconic of ABC which is

tangent to the sides of ABC at D, E, F, and let T_1 be the unique

affine map taking ABC to DEF. We have proved the following theorem.

Generalized Feuerbach Theorem. If P does not lie on a median of ABC,

the 9-point conic N_H is tangent to the conic I_P at the point Z which

is the center of the conic C = ABCPQ.

We prove this theorem by showing that the map M = T_1 * K^-1 * T_2 *

K^-1 takes the conic N_H to I_P and fixes the point Z. This map M can

be broken down as follows:

1) The anti-complement map K^-1 takes the conic N_H to the

circumconic C_O, whose center is of course O.

2) The map T_2 takes the circumconic C_O to the 9-point conic

N_P', whose center is K(Q).

3) The map K^-1 takes the conic N_P' to a circumconic of ABC with

center Q, which coincides with the inconic I_Q' of the anticevian

triangle of the point Q', defined as the isotomcomplement of the point

P' = complement of P. The anticevian triangle of Q' with respect to

ABC is the triangle (T_1)^-1(ABC), and the conic

K^-1(N_P') is tangent to the sides of this triangle at A, B, C because

of the general fact that Q is the isotomcomplement of Q' with respect

to triangle (T_1)^-1(ABC).

4) Grinberg (Hyacinthos #6423) has shown that the center of the

conic I_P is Q, and Ehrmann (Hyacinthos #959) noticed that Q is a

fixed point of the map T_1. Further, we have proved that T_1(Q') = P.

Thus, the map T_1 takes the conic I_Q' of step 3 to the inconic I_P.

This shows that M takes the 9-point conic N_H to the conic I_P. Now

the maps T_1 and T_2 are inverse maps on the line at infinity, so M is

a homothety or translation. We use the representation Z = GV.T_1(GV),

where V is the reflection of P in the point Q and G is the centroid,

to show that Z is the center of the mapping M and is therefore a fixed

point. Finally, Z lies on the 9-point conic N_H because Z is the

center of one of the conics lying on the quadrangle ABCH. Since M

fixes lines through Z, the tangent to N_H at Z coincides with the

tangent to I_P at Z, and this implies the theorem as stated above.

This gives a new proof of the Feuerbach's theorem in the special case

that P is the Gergonne point of triangle ABC, since in that case Q is

the incenter of ABC and H is the ordinary orthocenter. It turns out

that four different points P have the same generalized orthocenter, so

the theorem implies that four different inscribed conics are tangent

to the 9-point conic of ABCH. Thus, we get the full content of

Feuerbach's theorem in the classical case.

The full (synthetic) proofs can be found in our PDF manuscript in the

IUPUI Math Department Preprint Series at the website

http://www.math.iupui.edu/preprint/2009/ .

Igor Minevich and Patrick Morton

Dept. of Mathematical Sciences

Indiana University - Purdue University at Indianapolis (IUPUI)

pmorton@... - Dear Igor

If I understand you, you give the affine plane a metric structure, in

order to P would be the Gergonne point, P' the Nagel point and Q the center

of the incircle.

So with this new metrix , you have just to write again the old theory?

Friendly

Francois

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

We work in the extended Euclidean plane, and use Euclidean, affine,

and projective arguments. If we understand your question correctly,

you are suggesting that we could derive our whole theory by applying

affine maps to the classical theory. This is not the case, because

for some points P the inconic is a hyperbola, as P is allowed to lie

outside of the triangle ABC, and there is no affine map that will take

such an inconic to the incircle of some other triangle.

Whenever the inconic is an ellipse, the induced involution of

conjugate points on the line at infinity is elliptic. In this case

the theory does follow from classical Euclidean results a la Coxeter

(in his "The Real Projective Plane") , except that along the way we

prove some results we believe to be new. When the inconic is a

hyperbola, the induced involution on the line at infinity is

hyperbolic, and the results for this case cannot be proven directly

from the old Euclidean results as far as we know.

The case P = X(7) = Gergonne point is one interesting case, but there

are many others. If P = X(4) = the orthocenter, the point Z is the

center of the Jerabek hyperbola, and our results show that a Feuerbach

theorem exists for the inconic of triangle ABC whose center is the

symmedian point X(6). This inconic is tangent at Z to the nine-point

conic of ABCH, where the generalized orthocenter H = X(66), the

isogonal conjugate of the Exeter point X(22). We do not know if this

result was noticed before.

We would be interested in knowing if anyone has used an affine map

like our map M = T_1*K^-1*T_2*K^-1 to prove the classical Feuerbach

theorem.

With friendly regards,

Igor Minevich and Patrick Morton

--- In Hyacinthos@yahoogroups.com, Francois Rideau

<francois.rideau@...> wrote:>

center

> Dear Igor

> If I understand you, you give the affine plane a metric structure, in

> order to P would be the Gergonne point, P' the Nagel point and Q the

> of the incircle.

> So with this new metrix , you have just to write again the old theory?

> Friendly

> Francois

>

>

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

> - Dear Igor and Dear Patrick

I was only joking for I knew that point P was allowed to go everywhere it

could inside as well outside the reference triangle ABC.

Whenever you start with an affine plane , a reference triangle ABC in it

and some point P as a parameter, we can always ask if there exists an

euclidian structure on the plane in order to P becomes the orthocenter, or

the circumcenter, or the Gergonne point or any other point in the Kimberling

list.

Doing so we get an affine partition of the plane in two sets, in the first

there exists such an euclidian metric given by some quadratic form with a

(2,0) signature and in the other there exists also such a metric but

associated to a quadratic form with a (1, 1) signature and we have to work

in a hyperbolic plane.

For example, in the case of the orthocenter, the partition is given by the

side lines, for the circumcenter it is given by the sides of the medial

triangle, for the Lemoine point by the inner Steiner ellipse and so on...

I notice that in some cases, euclidian properties of the point P are still

valid provided they are translated properly, for example euclidian circles

becoming "circles" in the hyperbolic geometry, that is to say hyperbolas of

which asymptots are isotropic lines, and so on..

Maybe Poncelet should have been happy to see his continuity principle in

action!

I think doing geometry in a hyperbolic plane is very funny and interesting

and I will look at your paper with great pleasure.

Very friendly regards

Francois Rideau

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