## Feuerbach and pedal triangles

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• Dear Hyacinthians, ... The perpendiculars from each excenter to the line joining the Feuerbach point with the corresponding vertex of the medial triangle are
Message 1 of 3 , Dec 1, 2004
Dear Hyacinthians,

in message 8533 I wrote:
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The perpendiculars from each excenter to the line joining the
Feuerbach point with the corresponding vertex of the medial triangle
are concurrent.
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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, ... Your conjecture is correct. The locus of the perspector is the Jerabek hyperbola of the excentral triangle. With reference to ABC, this has
Message 2 of 3 , Dec 1, 2004
Dear Eric,

[ED]: 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 ?

Your conjecture is correct. The locus of the perspector is the Jerabek
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, ... Apart from the OI line, there is also the line containing X(36) and its isogonal conjugate X(80). X(36) is the inversive image of I in the
Message 3 of 3 , Dec 1, 2004
Dear Eric,

[ED]: 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?

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