Radical Axes and Loci

Let ABC be a triangle, L a line and A',B',C' the orthogonal projections of A,B,C on L.

Denote:

R1 = the radical axis of (B, BB'), (C, CC')

[ie the radical axis of the circles centered at B,C with radii BB',CC', resp.]

R2 = the radical axis of (C,CC'), (A,AA')

R3 = the radical axis of (A,AA'), (B,BB')

R1, R2, R3 are concurrent at a point R(L) = the Radical Center of the Circles (A,AA'), (B,BB'), (C, CC').

Loci:

1. Let P be a point and L a line passing through P.

Which is the locus of R(L) points as L moves around P?

2. Let P be a point on the circumcircle and L the Simson line of P.

Which is the locus of R(L) as P moves on the circumcircle?

3. Let P be a point and La,Lb,Lc three lines passing trrough P ( i) parallels or (ii) perpendiculars to BC,CA,AB, resp.

Which is the locus of P such that the triangles ABC, R(La)R(Lb)R(Lc) are perspective?

Antreas P. Hatzipolakis, Hyacinthos #21815

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1. It is an ellipse centered at the midpoint of segment OP. The ellipse degenerates when P is on NPC.

2. It is the Steiner deltoid of the medial triangle.

3 (i) A hyperbola similar to Jerabek hyperbola, but through O and and the nine point center. It intersects the Jerabek hyperbola at the isogonal conjugate of X3523.

(ii) It is the Darboux cubic of the medial triangle.

Francisco Javier, Hyacinthos #21819

The ratio squared of the homothety that carries Jerabek hyperbola into the hyperbola in 3(i) is

(p^2 - r^2 - 4 r R + 8 R^2)/(4 (p - r - 2 R) (p + r + 2 R)).

Francisco Javier, Hyacinthos #21826

Equation of hyperbola 3(i):

5 a^4 b^4 c^2 x^2 - 6 a^2 b^6 c^2 x^2 + b^8 c^2 x^2 - 5 a^4 b^2 c^4 x^2 - 3 b^6 c^4 x^2 + 6 a^2 b^2 c^6 x^2 + 3 b^4 c^6 x^2 - b^2 c^8 x^2 + a^8 c^2 x y + 2 a^6 b^2 c^2 x y - 2 a^2 b^6 c^2 x y - b^8 c^2 x y - 3 a^6 c^4 x y - 3 a^4 b^2 c^4 x y + 3 a^2 b^4 c^4 x y + 3 b^6 c^4 x y + 3 a^4 c^6 x y - 3 b^4 c^6 x y - a^2 c^8 x y + b^2 c^8 x y - a^8 c^2 y^2 + 6 a^6 b^2 c^2 y^2 - 5 a^4 b^4 c^2 y^2 + 3 a^6 c^4 y^2 + 5 a^2 b^4 c^4 y^2 - 3 a^4 c^6 y^2 - 6 a^2 b^2 c^6 y^2 + a^2 c^8 y^2 - a^8 b^2 x z + 3 a^6 b^4 x z - 3 a^4 b^6 x z + a^2 b^8 x z - 2 a^6 b^2 c^2 x z + 3 a^4 b^4 c^2 x z - b^8 c^2 x z - 3 a^2 b^4 c^4 x z + 3 b^6 c^4 x z + 2 a^2 b^2 c^6 x z - 3 b^4 c^6 x z + b^2 c^8 x z - a^8 b^2 y z + 3 a^6 b^4 y z - 3 a^4 b^6 y z + a^2 b^8 y z + a^8 c^2 y z - 3 a^4 b^4 c^2 y z + 2 a^2 b^6 c^2 y z - 3 a^6 c^4 y z + 3 a^4 b^2 c^4 y z + 3 a^4 c^6 y z - 2 a^2 b^2 c^6 y z - a^2 c^8 y z + a^8 b^2 z^2 - 3 a^6 b^4 z^2 + 3 a^4 b^6 z^2 - a^2 b^8 z^2 - 6 a^6 b^2 c^2 z^2 + 6 a^2 b^6 c^2 z^2 + 5 a^4 b^2 c^4 z^2 - 5 a^2 b^4 c^4 z^2

Francisco Javier

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