- Dear Paul

> [APH]: Let ABC be a triangle. If BC is fixed, which is the locus of A so

Construct an equilateral triangle HBC having base BC. Let A be its

> that the NPC center of ABC lies on AB ?

>

> ***

> [PY]: This is the rectangular hyperbola with B and C as vertices.

orthocenter. The four triangles ABC, HBC, HAB, HAC have the same NPC center

: A, the orthocenter of the equilateral triangle HBC, which lies on the

perpendicular bisector of BC, but not on the rectangular hyperbola with B

and C as vertices.

Friendly

Gilles - Dear Antreas, Paul and other Hyacinthists

> [APH]: Let ABC be a triangle. If BC is fixed, which is the locus of A so

Let H be the orthocenter of ABC: the four triangles ABC, HAB, HBC,HCA have

> that the NPC center of ABC lies on AB ?

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 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