ADGEOM 4649 * ADGEOM 4650

#4649

Dear Geometers,

1. Let DEF be cevian triangle of Nagels point Na. Ha,Hb,Hc be orthocenters of NaEF, NaFD, NaDE respectively. Perpendiculars from Ha to BC, Hb to CA and Hc to AB concur. (ıts a construction for X(3057).

2. Let DEF be cevian triangle of Nagels point Na. Ga,Gb,Gc be centroids of NaEF, NaFD, NaDE respectively. Perpendiculars from Ga to BC, Gb to CA and Gc to AB concur. (ıts a not listed in ETC).

Best regards

Abdilkadir Altıntaş

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#4650

[Abdilkadir Altıntas]:

2. Let DEF be cevian triangle of Nagels point Na. Ga,Gb,Gc be centroids
of NaEF, NaFD, NaDE respectively. Perpendiculars from Ga to BC, Gb to CA
and Gc to AB concur. (ıts a not listed in ETC).

**** The perpendiculars from Ga to BC, Gb to CA and Gc to AB concur at

W = (r-2 R) O - (r+4 R)Na

with first barycentric coordinate:

a (-a^4 (b-c)^2+a^5 (b+c)-(b^2-c^2)^2 (b^2+4 b c+c^2)-2 a^3 (b^3+2 b^2
c+2 b c^2+c^3)+a (b-c)^2 (b^3+5 b^2 c+5 b c^2+c^3)+2 a^2 (b^4+b^3 c+4
b^2 c^2+b c^3+c^4))

W is the reflection of X(i) in X(j), for these {i, j}: {354,10175},
{3576,3740}, {3753,5790}, {3892,10171}, {5603,10157}, {10164,3956}.

W lies on lines X(i)X(j) for these {i, j}: {3,3697}, {4,8}, {5,3555},
{10,1071}, {12,354}, {40,4662}, {84,165}, {200,1012}, {210,515},
{392,952}, {405,5534}, {495,5728}, {518,5587}, {519,15064}, {912,3753},
{936,958}, {942,5261}, {944,5044}, {956,5720}, {960,5881}, {971,5657},
{997,18236}, {1385,5260}, {1532,4847}, {1698,12675}, {1737,17625},
{2836,13214}, {3036,12665}, {3086,17624}, {3090,5045}, {3295,9844},
{3617,12528}, {3621,13600}, {3678,14110}, {3679,6001}, {3698,5884},
{3870,6913}, {3889,5056}, {3892,10171}, {3916,11499}, {3921,10167},
{3935,6912}, {3956,10164}, {3983,6684}, {4015,4297}, {4533,18525},
{4882,12705}, {5049,11374}, {5173,10590}, {5251,5531}, {5290,5902},
{5302,10902}, {5439,9956}, {5603,10157}, {5687,7330}, {5693,5836},
{5726,18412}, {5770,17612}, {5780,10246}, {5791,10786}, {5886,11240},
{5904,7686}, {5919,10950}, {6927,12125}, {6965,18527}, {7580,18528},
{7989,13374}, {8164,11018}, {8168,12703}, {8728,10202}, {9004,12587},
{9708,18446}, {9780,9940}, {10039,12711}, {10176,18250}, {10573,12709},
{10588,16193}, {11362,12688}, {12599,12692}, {12629,12635}

(6 - 9 - 13) - search numbers of W: (1.49964716737654,
-0.605533359430132, 3.36773503810838)

Angel Montesdeoca