HYACINTHOS 28651

[Tran Quang Hung]:

 

Let ABC be a triangle with NPC center N.

The reflection of the circle (NBC) in the line BC meets CA, AB again at Ac, Ab, resp.

Define similarly the points Ba, Bc, Cb, Ca.

Let Oa, Ob, Oc be the circumcenters of the triangles AAcAb, BBaBc, CCbCa, resp.

Then the NPC center of the triangle OaObOc lies on the Euler line of ABC.
Which is this point?

Let Ka, Kb, Kc be the circumcenters of the triangles ACaBa, BCbAb, CAcBc, resp.

Then the orthocenter of the triangle KaKbKc lies on the Euler line of ABC.
Which is this point?


[Peter Moses]:

Hi Antreas,

> Then the NPC center of the triangle OaObOc lies on the Euler line of ABC. 

Which is this point?  

X(20030)

> Then the orthocenter of the triangle KaKbKc lies on the Euler line of ABC. Which is this point?

2 a^16-11 a^14 b^2+25 a^12 b^4-29 a^10 b^6+15 a^8 b^8+3 a^6 b^10-9 a^4 b^12+5 a^2 b^14-b^16-11 a^14 c^2+34 a^12 b^2 c^2-35 a^10 b^4 c^2+12 a^8 b^6 c^2-9 a^6 b^8 c^2+26 a^4 b^10 c^2-25 a^2 b^12 c^2+8 b^14 c^2+25 a^12 c^4-35 a^10 b^2 c^4+12 a^8 b^4 c^4-3 a^6 b^6 c^4-16 a^4 b^8 c^4+45 a^2 b^10 c^4-28 b^12 c^4-29 a^10 c^6+12 a^8 b^2 c^6-3 a^6 b^4 c^6-2 a^4 b^6 c^6-25 a^2 b^8 c^6+56 b^10 c^6+15 a^8 c^8-9 a^6 b^2 c^8-16 a^4 b^4 c^8-25 a^2 b^6 c^8-70 b^8 c^8+3 a^6 c^10+26 a^4 b^2 c^10+45 a^2 b^4 c^10+56 b^6 c^10-9 a^4 c^12-25 a^2 b^2 c^12-28 b^4 c^12+5 a^2 c^14+8 b^2 c^14-c^16,-a^16+5 a^14 b^2-9 a^12 b^4+3 a^10 b^6+15 a^8 b^8-29 a^6 b^10+25 a^4 b^12-11 a^2 b^14+2 b^16+8 a^14 c^2-25 a^12 b^2 c^2+26 a^10 b^4 c^2-9 a^8 b^6 c^2+12 a^6 b^8 c^2-35 a^4 b^10 c^2+34 a^2 b^12 c^2-11 b^14 c^2-28 a^12 c^4+45 a^10 b^2 c^4-16 a^8 b^4 c^4-3 a^6 b^6 c^4+12 a^4 b^8 c^4-35 a^2 b^10 c^4+25 b^12 c^4+56 a^10 c^6-25 a^8 b^2 c^6-2 a^6 b^4 c^6-3 a^4 b^6 c^6+12 a^2 b^8 c^6-29 b^10 c^6-70 a^8 c^8-25 a^6 b^2 c^8-16 a^4 b^4 c^8-9 a^2 b^6 c^8+15 b^8 c^8+56 a^6 c^10+45 a^4 b^2 c^10+26 a^2 b^4 c^10+3 b^6 c^10-28 a^4 c^12-25 a^2 b^2 c^12-9 b^4 c^12+8 a^2 c^14+5 b^2 c^14-c^16 : : 

 

= lies on these lines: {2,3}, {195,11671}, {930,24573}, {1263,25044}, {6343,20424}, {10627,20327}, {15345,20414}

= reflection of X(i) in X(j) for these {i,j}: {3,10285}, {4,20120}, {20,14142}, {10205,5501}, {10627,20327}, {15345,20414}, {27868,20030}

Best regards,
Peter Moses.