HYACINTHOS 25316

 

[Antreas P. Hatzipolakis]:

 

 

1. Let ABC be a triangle.

Denote:

A'B'C' = the midway triangle of O (ie A', B', C' = the midpoints of AO,BO,CO, resp.)

A"B"C" = the reflection triangle of A'B'C' (ie A",B",C"= the reflections of A',B',C' in B'C', C'A', A'B', resp.)

The circumcircles of AOA", BOB", COC" are coaxial.

2. Variation (taking A'B'C' as reference triangle)

Let ABC be a triangle.

Denote:

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

A"B"C" = the reflection triangle of ABC (ie A",B",C" = the reflections of A,B,C in BC,CA, AB, resp.)

The circumcircles of A'OA", B'OB", C'OC" are coaxial.

Second points of intersections?

[Peter Moses]:

Hi Antreas,

 

1) a^2 (2 a^10-9 a^8 b^2+16 a^6 b^4-14 a^4 b^6+6 a^2 b^8-b^10-9 a^8 c^2+16 a^6 b^2 c^2-3 a^4 b^4 c^2-7 a^2 b^6 c^2+3 b^8 c^2+16 a^6 c^4-3 a^4 b^2 c^4+4 a^2 b^4 c^4-2 b^6 c^4-14 a^4 c^6-7 a^2 b^2 c^6-2 b^4 c^6+6 a^2 c^8+3 b^2 c^8-c^10):: = a^2 (SA+(S (R^2-2 S Cot[w]))/(6 S-R^2 Cot[w]))
on line {3,6}

 

 

2) a^2 (a^10-5 a^8 b^2+9 a^6 b^4-7 a^4 b^6+2 a^2 b^8-5 a^8 c^2+9 a^6 b^2 c^2+a^4 b^4 c^2-6 a^2 b^6 c^2+b^8 c^2+9 a^6 c^4+a^4 b^2 c^4+2 a^2 b^4 c^4-b^6 c^4-7 a^4 c^6-6 a^2 b^2 c^6-b^4 c^6+2 a^2 c^8+b^2 c^8),::
on line {3,6}.

 

Best regards,

Peter Moses.