HYACINTHOS 26601

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and S a point.

The parallel through A to SC intersects BC at A1
The parallel through B to SA intersects CA at B1
The parallel through C to SB intersects AB at C1

Let L1, M1, N1 be the midpoints of AA1, BB1, CC1, resp.

The points A* = AN1 /\ B1C1, B* = BL1 /\ C1A1, C* = CM1 /\ A1B1 are collinear.

Source: Stathis Koutras, Romantics of Geometry, 1165a  

Similarly:

Let ABC be a triangle and S a point.

The parallel through A to SB intersects BC at A2
The parallel through B to SC intersects CA at B2
The parallel through C to SA intersects AB at C2

Let L2, M2, N2 be the midpoints of AA2, BB2, CC2, resp.

The points A** = AM2 /\ B2C2, B** = BN2 /\ C2A2, C** = CL2 /\ A2B2 are collinear.

Which point is the point of intersection of the lines A*B*C*, A**B**C** ?

Especially for S = a simple point: O,H, G, N, I.....


[Peter Moses]:


Hi Antreas,

For some point, S{p,q,r}, the intersection is
p^3 - q^3 - r^3 + 3 p^2 (q + r) - 2 q r (q + r) - p q r::
a point,
3 (p^3 + q^3 + r^3 + 2 (q r (q + r)+ r p (r + p) + p q (p + q) + 3 p q r)) G - (p + q + r) (2 (p^2 + q^2 + r^2) + 5 (q r + r p + p q)) S,
on the line GS.

I -> a^3+3 a^2 b-b^3+3 a^2 c-a b c-2 b^2 c-2 b c^2-c^3::
on lines {{1,2},{320,4938},{522,4813},. ..}.

O -> a^12-6 a^10 b^2+12 a^8 b^4-9 a^6 b^6+3 a^2 b^10-b^12-6 a^10 c^2+11 a^8 b^2 c^2-3 a^6 b^4 c^2+2 a^4 b^6 c^2-9 a^2 b^8 c^2+5 b^10 c^2+12 a^8 c^4-3 a^6 b^2 c^4-4 a^4 b^4 c^4+6 a^2 b^6 c^4-11 b^8 c^4-9 a^6 c^6+2 a^4 b^2 c^6+6 a^2 b^4 c^6+14 b^6 c^6-9 a^2 b^2 c^8-11 b^4 c^8+3 a^2 c^10+5 b^2 c^10-c^12::
on Euler line {2,3}

H -> 5 a^10 b^2-15 a^8 b^4+14 a^6 b^6-2 a^4 b^8-3 a^2 b^10+b^12+5 a^10 c^2-4 a^8 b^2 c^2-2 a^6 b^4 c^2-8 a^4 b^6 c^2+13 a^2 b^8 c^2-4 b^10 c^2-15 a^8 c^4-2 a^6 b^2 c^4+20 a^4 b^4 c^4-10 a^2 b^6 c^4+7 b^8 c^4+14 a^6 c^6-8 a^4 b^2 c^6-10 a^2 b^4 c^6-8 b^6 c^6-2 a^4 c^8+13 a^2 b^2 c^8+7 b^4 c^8-3 a^2 c^10-4 b^2 c^10+c^12:: 

on Euler line {2,3}.

 

N -> 6 a^12-28 a^10 b^2+48 a^8 b^4-33 a^6 b^6+a^4 b^8+9 a^2 b^10-3 b^12-28 a^10 c^2+54 a^8 b^2 c^2-17 a^6 b^4 c^2+2 a^4 b^6 c^2-29 a^2 b^8 c^2+18 b^10 c^2+48 a^8 c^4-17 a^6 b^2 c^4-6 a^4 b^4 c^4+20 a^2 b^6 c^4-45 b^8 c^4-33 a^6 c^6+2 a^4 b^2 c^6+20 a^2 b^4 c^6+60 b^6 c^6+a^4 c^8-29 a^2 b^2 c^8-45 b^4 c^8+9 a^2 c^10+18 b^2 c^10-3 c^12:: 

on Euler line {2,3}

.
K -> a^6+3 a^4 b^2-b^6+3 a^4 c^2-a^2 b^2 c^2-2 b^4 c^2-2 b^2 c^4-c^6:: 

on lines {{2,6},{525,8665},...}.

X(8) -> X(5212).

X(10) -> 6 a^3+10 a^2 b-3 b^3+10 a^2 c-2 a b c-9 b^2 c-9 b c^2-3 c^3:: 

on lines {{1,2},...}.

Best regards,
Peter Moses.