HYACINTHOS 25344

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle, A'B'C' the pedal triangle of H and P a point.

Denote:

A",B",C" = the reflections of P in BC, CA, AB, resp.

A*, B*, C* = the reflections of A", B", C" in B'C', C'A', A'B', resp.

Ab, Ac = the orthogonal projections of A* on AC, AB, resp.

Bc, Ba = the orthogonal projections of B* on BA, BC, resp.

Ca, Cb = the orthogonal projections of C* on CB, CA, resp.

Oa, Ob, Oc = the circumcenters of A*AbAc, B*BcBa, C*CaCb, resp.

For P = O:

 

1. ABC, OaObOc are homothetic

2. A"B"C", OaObOc are homothetic

Which points are the homothetic centers (on the Euler line)?

3. ABC, OaObOc are orthologic.

The orthologic center (ABC, OaObOc) is the H.

Which point is the other one (OaObOc, ABC) [on the Euler line]?

Which is the locus of P such that ABC, OaObOc are
a. perspective
b. orthologic?

[Peter Moses]:

Hi Antreas,

 

1) X(26).

 

2) a^10-a^8 b^2-2 a^6 b^4+2 a^4 b^6+a^2 b^8-b^10-a^8 c^2-4 a^4 b^4 c^2+2 a^2 b^6 c^2+3 b^8 c^2-2 a^6 c^4-4 a^4 b^2 c^4-6 a^2 b^4 c^4-2 b^6 c^4+2 a^4 c^6+2 a^2 b^2 c^6-2 b^4 c^6+a^2 c^8+3 b^2 c^8-c^10::
on lines {{2,3},{68,143},...} Searches {-0.88466424430114870039903323 9615,-1.7518504005848276963283 5858389,5.26179056429748229306 831177117}.

Reflection of X(7514) in X(5).

 

3) 2 a^10-3 a^8 b^2-2 a^6 b^4+4 a^4 b^6-b^10-3 a^8 c^2-2 a^4 b^4 c^2+2 a^2 b^6 c^2+3 b^8 c^2-2 a^6 c^4-2 a^4 b^2 c^4-4 a^2 b^4 c^4-2 b^6 c^4+4 a^4 c^6+2 a^2 b^2 c^6-2 b^4 c^6+3 b^2 c^8-c^10::
on lines {{2,3},{51,11750},...}
Searches {-3.18469148443498600227301307 083,-4.04581011078426256326240 328836,7.911467551420400571041 41033483}.

Midpoint of X(i) and X(j) for these {i,j}: {{382,6240},{3575,7553}}.

Reflection of X(i) in X(j) for these {i,j}: {{5,6756},{1885,3853},{6146,14 3}}.

 

a) a circular cubic through {3,403,1658}.

b) line through{3,49,155,184,185,283,3 94,1092,1147,1181,1204,1216,14 37,1790,1800,1801,1819,3167,32 92,3796,3917,5406,5407,5408,54 09,5447,5562,7689,8913,9703,97 04,9720,9908,10132,10133,10605 ,10670,10674,10984}

& infinity.

 

Best regards,

Peter Moses.