## Re: [EMHL] NPC locus

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• Dear Antreas, Paul and other Hyacinthists ... Let H be the orthocenter of ABC: the four triangles ABC, HAB, HBC,HCA have the same NPC center N. A is the
Message 1 of 5 , Oct 3, 2001
Dear Antreas, Paul and other Hyacinthists

> [APH]: Let ABC be a triangle. If BC is fixed, which is the locus of A so
> that the NPC center of ABC lies on AB ?

Let H be the orthocenter of ABC: the four triangles ABC, HAB, HBC,HCA have
the same NPC center N.

A is the orthocenter of HBC, thus "N lies on AB" means "N lies on an
altitude of HBC".

N lies on an altitude iff this altitude is the euler line of the triangle
iff the triangle is isosceles.

N lies on AB iff BH=BC.

The locus of H is the circle with center B and radius BC.
The locus of A is the locus of the orthocenter of HBC :

an orthostrophoid with double point at B and asymptot perpendicular to BC at
C' reflection of C about B.

Friendly

Gilles
• Dear Gilles and Antreas, [APH]: Let ABC be a triangle. If BC is fixed, which is the locus of A so that the NPC center of ABC lies on AB ? ^^ [PY]: The locus is
Message 2 of 5 , Oct 3, 2001
Dear Gilles and Antreas,

[APH]: Let ABC be a triangle. If BC is fixed, which is the locus of A so that
the NPC center of ABC lies on AB ?
^^
[PY]: The locus is the rectangular hyperbola with vertices B and C.

[GB]: Let H be the orthocenter of ABC: the four triangles ABC, HAB, HBC,HCA
have the same NPC center N.

A is the orthocenter of HBC, thus "N lies on AB" means "N lies on an
altitude of HBC".

N lies on an altitude iff this altitude is the euler line of the triangle iff
the triangle is isosceles.

N lies on AB iff BH=BC.

The locus of H is the circle with center B and radius BC.
The locus of A is the locus of the orthocenter of HBC :

an orthostrophoid with double point at B and asymptot perpendicular to BC at
C' reflection of C about B.

****
Oh! I see. I had solved a different but easier problem: I thought Antreas was
asking for the locus of A for which the NPC of ABC lies on BC. Now I see he
wrote AB.

Best regards
Sincerely
Paul
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