HYACINTHOS 24042

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and P a point.

Denote:

Oa, Ob, Oc = the circumcenters of PBC,PCA, PAB, resp.

Oab, Oac = the orthogonal projections of Oa on AC, AB, resp.

O1 = the circumcenter of OaOabOac

Similarly O2 and O3.

1. P = O

The NPC center of O1O2O3 lies on the Euler line of ABC

Note: The circumcircles of OaOabOac, ObObcOba, OcOcaOcb and ABC

are concurrent at the Euler line reflection point.

2. P = N

The O (circumcenter) of O1O2O3 lies on the Euler line of ABC.

3. P = I

The circumcenter of O1O2O3 is the midpoint of OI.

4. P = H

The circumcircles of OaOabOac, ObObcOba, OcOcaOcb are concentric

with the NPC of ABC.

[Peter Moses]:

Hi Antreas,

 

1). 2 a^10-5 a^8 b^2+2 a^6 b^4+4 a^4 b^6-4 a^2 b^8+b^10-5 a^8 c^2+6 a^6 b^2 c^2-3 a^4 b^4 c^2+5 a^2 b^6 c^2-3 b^8 c^2+2 a^6 c^4-3 a^4 b^2 c^4-2 a^2 b^4 c^4+2 b^6 c^4+4 a^4 c^6+5 a^2 b^2 c^6+2 b^4 c^6-4 a^2 c^8-3 b^2 c^8+c^10:: on lines {{2,3},{125,5944},{5946,8254}} , Searches {2. 55522495183134755866305138624, 1. 67896428099220314957622289805, 1. 29897000191377035710571630251} .

 

2). 2 a^16-13 a^14 b^2+39 a^12 b^4-69 a^10 b^6+75 a^8 b^8-47 a^6 b^10+13 a^4 b^12+a^2 b^14-b^16-13 a^14 c^2+54 a^12 b^2 c^2-85 a^10 b^4 c^2+44 a^8 b^6 c^2+29 a^6 b^8 c^2-38 a^4 b^10 c^2+5 a^2 b^12 c^2+4 b^14 c^2+39 a^12 c^4-85 a^10 b^2 c^4+44 a^8 b^4 c^4+9 a^6 b^6 c^4+18 a^4 b^8 c^4-21 a^2 b^10 c^4-4 b^12 c^4-69 a^10 c^6+44 a^8 b^2 c^6+9 a^6 b^4 c^6+14 a^4 b^6 c^6+15 a^2 b^8 c^6-4 b^10 c^6+75 a^8 c^8+29 a^6 b^2 c^8+18 a^4 b^4 c^8+15 a^2 b^6 c^8+10 b^8 c^8-47 a^6 c^10-38 a^4 b^2 c^10-21 a^2 b^4 c^10-4 b^6 c^10+13 a^4 c^12+5 a^2 b^2 c^12-4 b^4 c^12+a^2 c^14+4 b^2 c^14-c^16:: on lines {{2,3},{1209,6592},{6150,8254} } Searches {7. 40818277605552391582959611165, 6. 51911987994049134601159528011, -4. 29173363930774132834948521987}

 

Best regards

Peter Moses.