- Dear Hyacinthians,

in message 8533 I wrote:

-----------------------------------------------------------------

The perpendiculars from each excenter to the line joining the

Feuerbach point with the corresponding vertex of the medial triangle

are concurrent.

-----------------------------------------------------------------

I conjecture the following generalization:

Let O be the circumcenter and I the incenter

The perpendiculars from each excenter to the line joining the

Feuerbach point with the corresponding vertex of the pedal triangle

of any point on OI are concurrent.

The following questions intrigue me:

==> what is the locus of the perspectors ?

==> is OI the only set of points with this property?

==> if we replace the Feuerbach point by another point is there

another line, or is this property unique for the Feuerbach point ?

I suppose it all has something to do with orthopoles since the

Feuerbach point is the orthopole of the OI-line, but I can't find

out why

Greetings from Bruges

Eric - Dear Eric,

[ED]: I conjecture the following generalization:

>Let O be the circumcenter and I the incenter

Your conjecture is correct. The locus of the perspector is the Jerabek

>The perpendiculars from each excenter to the line joining the

>Feuerbach point with the corresponding vertex of the pedal triangle

>of any point on OI are concurrent.

>

>The following questions intrigue me:

>

>==> what is the locus of the perspectors ?

hyperbola of the excentral triangle.

With reference to ABC, this has equation

bc(b-c)x^2 + ca(c-a)y^2 + ab(a-b)z^2 = 0.

Best regards

Sincerely

Paul

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

[ED]: I conjecture the following generalization:

>Let O be the circumcenter and I the incenter

Apart from the OI line, there is also the line containing X(36) and its

>The perpendiculars from each excenter to the line joining the

>Feuerbach point with the corresponding vertex of the pedal triangle

>of any point on OI are concurrent.

>

>The following questions intrigue me:

>

>==> what is the locus of the perspectors ?

>==> is OI the only set of points with this property?

isogonal conjugate X(80).

X(36) is the inversive image of I in the circumcircle.

Best regards

Sincerely

Paul

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