INCIRCLE

CIRCUMCIRCLE

A circumcircle with thousands of inscribed memories....

Perspective

Let ABC be a triangle, A1B1C1 the orthic triangle, A2B2C2 the circumcevian triangle of O with respect A1B1C1, A3B3C3 the antipodal triangle of A2B2C2 and A'B'C' the triangle bounded by the lines A1A3, B1B3, C1C3, resp..

The triangles A1B1C1 and A'B'C' are perspective.

Perspector?

APH, 19 March 2012

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The perspector is X25.

Francisco Javier García Capitán
19 March 2012

HOW THEY DO GEOMETRY IN CHINA

how they do geometry in China

Perspective

Let ABC be a triangle, A'B'C' the cevian triangle of G and A"B"C" the circumcevian triangle of G with respect A'B'C'. The circles with diameters HA",HB",HC" intersect the NPC again at A1,B1,C1, resp.

The circles with diameters HA",HB",HC" intersect the NPC again at A1,B1,C1, resp.

The triangles ABC, A1B1C1 are perspective.

Perspector?

APH, 16 March 2012

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Not in ETC. It is the isotomic conjugate of

(SA (b^4 + c^4-a^4), SB (a^4 - b^4 + c^4), SC (a^4 + b^4 - c^4))

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ADDENDUM (12/9/19)
Perspector: X(13854)
Isotomic conjugate: X(34254)
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The locus of P, instead of H, is trilinear polar of X648 (containing H) + NPC + a cubic:

5 a^12 x^3 - 28 a^10 b^2 x^3 + 53 a^8 b^4 x^3 - 40 a^6 b^6 x^3 +
7 a^4 b^8 x^3 + 4 a^2 b^10 x^3 - b^12 x^3 - 28 a^10 c^2 x^3 +
110 a^8 b^2 c^2 x^3 - 120 a^6 b^4 c^2 x^3 + 36 a^4 b^6 c^2 x^3 +
4 a^2 b^8 c^2 x^3 - 2 b^10 c^2 x^3 + 53 a^8 c^4 x^3 -
120 a^6 b^2 c^4 x^3 + 42 a^4 b^4 c^4 x^3 - 8 a^2 b^6 c^4 x^3 +
b^8 c^4 x^3 - 40 a^6 c^6 x^3 + 36 a^4 b^2 c^6 x^3 -
8 a^2 b^4 c^6 x^3 + 4 b^6 c^6 x^3 + 7 a^4 c^8 x^3 +
4 a^2 b^2 c^8 x^3 + b^4 c^8 x^3 + 4 a^2 c^10 x^3 - 2 b^2 c^10 x^3 -
c^12 x^3 + a^12 x^2 y - 4 a^10 b^2 x^2 y - 7 a^8 b^4 x^2 y +
40 a^6 b^6 x^2 y - 53 a^4 b^8 x^2 y + 28 a^2 b^10 x^2 y -
5 b^12 x^2 y - 2 a^10 c^2 x^2 y + 24 a^8 b^2 c^2 x^2 y -
44 a^6 b^4 c^2 x^2 y + 48 a^4 b^6 c^2 x^2 y - 34 a^2 b^8 c^2 x^2 y +
8 b^10 c^2 x^2 y - 17 a^8 c^4 x^2 y + 46 a^4 b^4 c^4 x^2 y +
3 b^8 c^4 x^2 y + 36 a^6 c^6 x^2 y - 24 a^4 b^2 c^6 x^2 y +
4 a^2 b^4 c^6 x^2 y - 8 b^6 c^6 x^2 y - 17 a^4 c^8 x^2 y +
4 a^2 b^2 c^8 x^2 y + b^4 c^8 x^2 y - 2 a^2 c^10 x^2 y +
c^12 x^2 y - 5 a^12 x y^2 + 28 a^10 b^2 x y^2 - 53 a^8 b^4 x y^2 +
40 a^6 b^6 x y^2 - 7 a^4 b^8 x y^2 - 4 a^2 b^10 x y^2 + b^12 x y^2 +
8 a^10 c^2 x y^2 - 34 a^8 b^2 c^2 x y^2 + 48 a^6 b^4 c^2 x y^2 -
44 a^4 b^6 c^2 x y^2 + 24 a^2 b^8 c^2 x y^2 - 2 b^10 c^2 x y^2 +
3 a^8 c^4 x y^2 + 46 a^4 b^4 c^4 x y^2 - 17 b^8 c^4 x y^2 -
8 a^6 c^6 x y^2 + 4 a^4 b^2 c^6 x y^2 - 24 a^2 b^4 c^6 x y^2 +
36 b^6 c^6 x y^2 + a^4 c^8 x y^2 + 4 a^2 b^2 c^8 x y^2 -
17 b^4 c^8 x y^2 - 2 b^2 c^10 x y^2 + c^12 x y^2 - a^12 y^3 +
4 a^10 b^2 y^3 + 7 a^8 b^4 y^3 - 40 a^6 b^6 y^3 + 53 a^4 b^8 y^3 -
28 a^2 b^10 y^3 + 5 b^12 y^3 - 2 a^10 c^2 y^3 + 4 a^8 b^2 c^2 y^3 +
36 a^6 b^4 c^2 y^3 - 120 a^4 b^6 c^2 y^3 + 110 a^2 b^8 c^2 y^3 -
28 b^10 c^2 y^3 + a^8 c^4 y^3 - 8 a^6 b^2 c^4 y^3 +
42 a^4 b^4 c^4 y^3 - 120 a^2 b^6 c^4 y^3 + 53 b^8 c^4 y^3 +
4 a^6 c^6 y^3 - 8 a^4 b^2 c^6 y^3 + 36 a^2 b^4 c^6 y^3 -
40 b^6 c^6 y^3 + a^4 c^8 y^3 + 4 a^2 b^2 c^8 y^3 + 7 b^4 c^8 y^3 -
2 a^2 c^10 y^3 + 4 b^2 c^10 y^3 - c^12 y^3 + a^12 x^2 z -
2 a^10 b^2 x^2 z - 17 a^8 b^4 x^2 z + 36 a^6 b^6 x^2 z -
17 a^4 b^8 x^2 z - 2 a^2 b^10 x^2 z + b^12 x^2 z -
4 a^10 c^2 x^2 z + 24 a^8 b^2 c^2 x^2 z - 24 a^4 b^6 c^2 x^2 z +
4 a^2 b^8 c^2 x^2 z - 7 a^8 c^4 x^2 z - 44 a^6 b^2 c^4 x^2 z +
46 a^4 b^4 c^4 x^2 z + 4 a^2 b^6 c^4 x^2 z + b^8 c^4 x^2 z +
40 a^6 c^6 x^2 z + 48 a^4 b^2 c^6 x^2 z - 8 b^6 c^6 x^2 z -
53 a^4 c^8 x^2 z - 34 a^2 b^2 c^8 x^2 z + 3 b^4 c^8 x^2 z +
28 a^2 c^10 x^2 z + 8 b^2 c^10 x^2 z - 5 c^12 x^2 z +
10 a^12 x y z - 36 a^10 b^2 x y z + 22 a^8 b^4 x y z +
8 a^6 b^6 x y z + 22 a^4 b^8 x y z - 36 a^2 b^10 x y z +
10 b^12 x y z - 36 a^10 c^2 x y z + 124 a^8 b^2 c^2 x y z -
88 a^6 b^4 c^2 x y z - 88 a^4 b^6 c^2 x y z +
124 a^2 b^8 c^2 x y z - 36 b^10 c^2 x y z + 22 a^8 c^4 x y z -
88 a^6 b^2 c^4 x y z + 132 a^4 b^4 c^4 x y z -
88 a^2 b^6 c^4 x y z + 22 b^8 c^4 x y z + 8 a^6 c^6 x y z -
88 a^4 b^2 c^6 x y z - 88 a^2 b^4 c^6 x y z + 8 b^6 c^6 x y z +
22 a^4 c^8 x y z + 124 a^2 b^2 c^8 x y z + 22 b^4 c^8 x y z -
36 a^2 c^10 x y z - 36 b^2 c^10 x y z + 10 c^12 x y z + a^12 y^2 z -
2 a^10 b^2 y^2 z - 17 a^8 b^4 y^2 z + 36 a^6 b^6 y^2 z -
17 a^4 b^8 y^2 z - 2 a^2 b^10 y^2 z + b^12 y^2 z +
4 a^8 b^2 c^2 y^2 z - 24 a^6 b^4 c^2 y^2 z + 24 a^2 b^8 c^2 y^2 z -
4 b^10 c^2 y^2 z + a^8 c^4 y^2 z + 4 a^6 b^2 c^4 y^2 z +
46 a^4 b^4 c^4 y^2 z - 44 a^2 b^6 c^4 y^2 z - 7 b^8 c^4 y^2 z -
8 a^6 c^6 y^2 z + 48 a^2 b^4 c^6 y^2 z + 40 b^6 c^6 y^2 z +
3 a^4 c^8 y^2 z - 34 a^2 b^2 c^8 y^2 z - 53 b^4 c^8 y^2 z +
8 a^2 c^10 y^2 z + 28 b^2 c^10 y^2 z - 5 c^12 y^2 z - 5 a^12 x z^2 +
8 a^10 b^2 x z^2 + 3 a^8 b^4 x z^2 - 8 a^6 b^6 x z^2 +
a^4 b^8 x z^2 + b^12 x z^2 + 28 a^10 c^2 x z^2 -
34 a^8 b^2 c^2 x z^2 + 4 a^4 b^6 c^2 x z^2 + 4 a^2 b^8 c^2 x z^2 -
2 b^10 c^2 x z^2 - 53 a^8 c^4 x z^2 + 48 a^6 b^2 c^4 x z^2 +
46 a^4 b^4 c^4 x z^2 - 24 a^2 b^6 c^4 x z^2 - 17 b^8 c^4 x z^2 +
40 a^6 c^6 x z^2 - 44 a^4 b^2 c^6 x z^2 + 36 b^6 c^6 x z^2 -
7 a^4 c^8 x z^2 + 24 a^2 b^2 c^8 x z^2 - 17 b^4 c^8 x z^2 -
4 a^2 c^10 x z^2 - 2 b^2 c^10 x z^2 + c^12 x z^2 + a^12 y z^2 +
a^8 b^4 y z^2 - 8 a^6 b^6 y z^2 + 3 a^4 b^8 y z^2 +
8 a^2 b^10 y z^2 - 5 b^12 y z^2 - 2 a^10 c^2 y z^2 +
4 a^8 b^2 c^2 y z^2 + 4 a^6 b^4 c^2 y z^2 - 34 a^2 b^8 c^2 y z^2 +
28 b^10 c^2 y z^2 - 17 a^8 c^4 y z^2 - 24 a^6 b^2 c^4 y z^2 +
46 a^4 b^4 c^4 y z^2 + 48 a^2 b^6 c^4 y z^2 - 53 b^8 c^4 y z^2 +
36 a^6 c^6 y z^2 - 44 a^2 b^4 c^6 y z^2 + 40 b^6 c^6 y z^2 -
17 a^4 c^8 y z^2 + 24 a^2 b^2 c^8 y z^2 - 7 b^4 c^8 y z^2 -
2 a^2 c^10 y z^2 - 4 b^2 c^10 y z^2 + c^12 y z^2 - a^12 z^3 -
2 a^10 b^2 z^3 + a^8 b^4 z^3 + 4 a^6 b^6 z^3 + a^4 b^8 z^3 -
2 a^2 b^10 z^3 - b^12 z^3 + 4 a^10 c^2 z^3 + 4 a^8 b^2 c^2 z^3 -
8 a^6 b^4 c^2 z^3 - 8 a^4 b^6 c^2 z^3 + 4 a^2 b^8 c^2 z^3 +
4 b^10 c^2 z^3 + 7 a^8 c^4 z^3 + 36 a^6 b^2 c^4 z^3 +
42 a^4 b^4 c^4 z^3 + 36 a^2 b^6 c^4 z^3 + 7 b^8 c^4 z^3 -
40 a^6 c^6 z^3 - 120 a^4 b^2 c^6 z^3 - 120 a^2 b^4 c^6 z^3 -
40 b^6 c^6 z^3 + 53 a^4 c^8 z^3 + 110 a^2 b^2 c^8 z^3 +
53 b^4 c^8 z^3 - 28 a^2 c^10 z^3 - 28 b^2 c^10 z^3 + 5 c^12 z^3

Francisco Javier García Capitán
17 March 2012

Concyclic Os

Let ABC be a triangle and P a point.

For which P's the circumcenters O,Oa,Ob,Oc of ABC, PBC, PCA, PAB, resp. are concyclic?

APH, 28 February 2012

Oa,Ob,Oc

Let ABC be a triangle and A'B'C' the cevian triangle of I.

The reflection of BB' in AA' intersects AB in Ab
The reflection of CC' in AA' intersects AC in Ac

Let Oa be the circumcenter of A'AbAc

A'AbAc is the reflection of A'B'C' in AA' and Oa is the reflection
of the circumcenter of A'B'C' in AA'.

Similarly Ob, Oc.

Oa, Ob, Oc are the reflections of the circumcenter of the cevian triangle of I in the bisectors AA',BB',CC', resp. Therefore the lines AOa, BOb, COc concur at the isogonal conjugate of the circumcenter of A'B'C'.

Coordinates ?

Generalization:
For P instead of I:

We have two problems:

1. The locus of P such that ABC, OaObOc are perspective. (Oa,Ob,Oc as defined above)
2. The locus of P such that ABC, QaQbQc are perspective, where Qa is the circumcenter of the triangle which is the reflection of A'B'C' in AA' (ie Qa is the reflection of the circumcenter of A'B'C' in AA') and similarly Qb,Qc

APH, 24 February 2012

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Coordinates:

(a (a^5 b + a^4 b^2 - 2 a^3 b^3 - 2 a^2 b^4 + a b^5 + b^6 + a^5 c -
3 a^3 b^2 c - 2 a^2 b^3 c + 2 a b^4 c + 2 b^5 c + a^4 c^2 -
3 a^3 b c^2 - 2 a^2 b^2 c^2 - 3 a b^3 c^2 - b^4 c^2 - 2 a^3 c^3 -
2 a^2 b c^3 - 3 a b^2 c^3 - 4 b^3 c^3 - 2 a^2 c^4 + 2 a b c^4 -
b^2 c^4 + a c^5 + 2 b c^5 + c^6) : ...:...)

1. The locus is a 18 degree curve.

2. The locus is a 24 degree curve.

Nikos Dergiades, Hyacinthos #20870