ADGEOM 4765 * ADGEOM 4766 * ADGEOM 4767

#4765

Dear Geometers,

Let Ge Gergonne point of ABC.

DEF cevian triangle of Ge.

Na, NPC of GeFE. Define Nb,Nc cyclically.

Perpendiculars from Na to BC, Nb to CA and Nc to AB concur. (Not listed in ETC).

If DEF is cevian triangle of Nagels point and Na,Nb,Nc are NPC s defined above Perpendiculars from Na to BC, Nb to CA and Nc to AB concur. (Not listed in ETC).

Abdilkadir Altıntaş, Emirdağ, Türkiye

Best regards...

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#4766

[Abdilkadir Altıntas]:

Let Ge Gergonne point of ABC.
DEF cevian triangle of Ge.
Na, NPC of GeFE. Define Nb,Nc cyclically.

Perpendiculars from Na to BC, Nb to CA and Nc to AB concur. (Not listed
in ETC).

**** Perpendiculars from Na to BC, Nb to CA and Nc to AB concur at
U = (2r+7R) X(7) - (2r+3 R) X(79).

U = a (-a^4 (b+c)+5 a^2 b c (b+c)+2 a^3 (b^2+c^2)-2 a (b-c)^2 (b^2+3 b
c+c^2)+(b-c)^2 (b^3+c^3)) : ... : ...

U is the midpoint of X(i) and X(j), for these {i, j}: {9,3874},
{10,15185}, {942,5572}, {5542,5728}.

U is the reflection of X(3678) in X(6666)

W lies on lines X(i)X(j) for these {i, j}: {1,1170}, {7,79}, {9,3874},
{10,15185}, {11,118}, {142,3841}, {390,5902}, {516,942},
{518,1125}, {758,1001}, {971,12005}, {997,3243}, {1699,12669},
{2550,5883}, {2802,14563}, {3059,5439}, {3086,11038}, {3305,3873},
{3336,7676}, {3338,7675}, {3670,4343}, {3754,5853}, {3826,3833},
{3889,8583}, {4312,7671}, {4349,14523}, {5425,12758}, {5708,11495},
{5732,10980}, {5762,6583}, {5903,8236}, {5904,18230}, {6684,16216},
{8232,10399}, {12755,16173}, {13751,13995}, {15008,15726},
{15299,18389}.

(6 - 9 - 13) - search numbers of U: (1.34655429409374,
1.43528118005218, 2.02552168305884)


[Abdilkadir Altıntaş]:
If DEF is cevian triangle of Nagels point and Na,Nb,Nc are NPC s defined
above Perpendiculars from Na to BC, Nb to CA and Nc to AB concur. (Not
listed in ETC).

**** Perpendiculars from Na to BC, Nb to CA and Nc to AB concur at
V = (2r+3R) X(8) - (2r-5R) X(13729).

V = a (a^5 (b+c)-a^4 (b+c)^2-(b^2-c^2)^2 (b^2+b c+c^2)+a^3 (-2 b^3+b^2
c+b c^2-2 c^3)+a (b-c)^2 (b^3+c^3)+a^2 (2 b^4+3 b^3 c-2 b^2 c^2+3 b
c^3+2 c^4)) : ... : ...

V is the midpoint of X(i) and X(j), for these {i, j}: {5,5694},
{10,5887}, {72,946},}, {355,3878}, {960,5777},}, {5693,5884} ,
{5882,14872}, {11362,12672}, {11715,12665}.

V is the reflection of X(i) in X(j), for these {i, j}: {3754,9956},
{3881,5901}, {5690,4015}, {5885,3628}, {6684,5044}, {12005,1125}.

V lies on lines X(i)X(j) for these {i, j}: {1,6920}, {2,5693},
{3,3647}, {4,5692}, {5,758}, {8,13729}, {9,1630}, {10,119},
{21,6326}, {40,3876}, {52,15049}, {65,3614}, {72,946}, {79,6901},
{140,2771}, {191,6905}, {210,11362}, {355,3878}, {392,5882},
{515,960}, {517,546}, {518,13464}, {631,15071}, {912,1125},
{936,1158}, {952,3884}, {956,12059}, {997,5450}, {1071,10165},
{1216,2392}, {1339,10247}, {1376,5780}, {1385,2801}, {1656,5883},
{1768,6940}, {1844,7551}, {1858,13411}, {1898,4304}, {1935,11700},
{2778,5893}, {2779,5907}, {2842,10170}, {3057,18908}, {3090,5902},
{3219,11012}, {3336,6946}, {3428,15650}, {3452,12616}, {3485,18397},
{3576,12528}, {3628,3833}, {3681,7982}, {3743,5396}, {3754,3838},
{3812,10172}, {3817,4067}, {3868,8227}, {3869,5587}, {3873,9624},
{3874,5886}, {3877,5881}, {3881,5901}, {3901,7988}, {3940,11496},
{3951,12704}, {4005,13865}, {4015,5690}, {4127,9955}, {4134,4301},
{4187,10265}, {4661,5734}, {5044,6001}, {5086,6246}, {5225,5697},
{5250,17857}, {5251,12691}, {5535,6915}, {5603,5904}, {5719,12564},
{5720,6796}, {5770,10200}, {5779,12114}, {5811,6256}, {5818,5903},
{5927,14110}, {6839,16125}, {6881,11263}, {6909,7701}, {6913,12635},
{6916,16127}, {6986,16132}, {7686,10157}, {10914,14740},
{11112,16120}, {11375,18389}, {11715,12665}, {12047,15556},
{12736,17606}.

(6 - 9 - 13) - search numbers of V: (1.83745805188466,
0.208007872308717, 2.64860146943927).

Angel Montesdeoca

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#4767

 

In Addition:

Let Ge Gergonne point of ABC.
DEF cevian triangle of Ge.
Na, NPC of GeFE. Define Nb,Nc cyclically.
*** The orthology center of ABC with respect to NaNbNc is X(1156) = isogonal conjugate of  X(1155).       (X(1155) = SCHRÖDER POINT).

If DEF is cevian triangle of Nagels point and Na,Nb,Nc are NPC s defined above, then:
 *** The orthology center of ABC with respect to NaNbNc is X(1320) = isogonal conjugate of X(1319).    (X(1319) = BEVAN-SCHRÖDER POINT).
 
 Angel Montesdeoca