HYACINTHOS 28731

[Antreas P. Hatzipolakis]:

 

A PROPERTY OF THE FEUERBACH POINT Fe = X(11)

From a locus problem I posted to Hyacinthos (28935)

Let ABC be a triangle, A1B1C1 the pedal triangle of the Incenter I, and A'B'C' the pedal triangle of the Feuerbach point Fe.

Denote:

R1, R2, R3 = the reflections of FeA', FeB', FeC' in B1C1, C1A1, A1B1, resp.

A*B*C* = the triangle bounded by R1, R2, R3.

The parallels to BC, CA, AB through A*, B*, C*, resp. concur at Fe.

 

APH, Romantics of Geometry, 2381

 

 

[Aris Pavlakis]:

 

The triangles A1B1C1, A*B*C* are perspective.
The perspector lies on the incircle.

Perspector Q ? 

 

Aris Pavlakis, Comment in Romantics of Geometry, 2381 

[Antreas P. Hatzipolakis]:

Denote:

Q = the perspector corresponding to Feuerbach point Fe and the pedal triangle of the incenter I
(lying on the incircle (I))
Extraversions:
Qa = the perspector corresponding to ex-Feuerbach point Fa and the pedal triangle of the excenter Ia
(lying on the excircle (Ia))
Qb = the perspector corresponding to ex-Feuerbach point Fb and the pedal triangle of the excenter Ib
(lying on the excircle (Ib))  
Qc = the perspector corresponding to ex-Feuerbach point Fc and the pedal triangle of the excenter Ic
(lying on the excircle (Ic))  

1. Are the triangles ABC, QaQbQc perspective?

2. Are Q, Qa, Qb, Qc concyclic?


[Peter Moses]:

Hi Antreas,

Q = X(1364).

 

Qa = {a^2 (b-c)^2 (a+b+c) (a^2-b^2-c^2)^2,-b^2 (a+b-c) (a+c)^2 (-a^2+b^2-c^2)^2,-(a+b)^2 c^2 (a-b+c) (a^2+b^2-c^2)^2}.

1).  Yes. Perspector:

 

= X(7066) 

a^2 (a+b-c) (a-b+c) (b+c)^2 (a^2-b^2-c^2)^2 : :  

 

= lies on these lines: {3,1794}, {9,19366}, {10,12}, {34,26893}, {40,1745}, {55,581}, {56,219}, {64,7074}, {71,73}, {78,296}, {185,212}, {201,1425}, {227,22276}, {255,1364}, {329,1118}, {388,26872}, {389,3074}, {394,7335}, {511,1935}, {517,1838}, {603,3917}, {756,7324}, {970,24310}, {1038,3781}, {1361,3869}, {1397,19762}, {1469,5227}, {1490,6254}, {1682,22134}, {1762,29958}, {1802,7114}, {1859,5777}, {1936,5907}, {2175,10831}, {2218,3271}, {2323,19365}, {3075,11793}, {3611,18673}, {3682,22341}, {3695,7068}, {5285,26888}, {6285,7070}, {6354,15443}, {7085,19349}, {7352,26921}, {7957,10374}, {12835,23150}, {14059,20764}, {18915,26939}, {21015,26955}, {23154,26934}

 

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {255, 5562, 1364}, {1425, 3690, 201}
= isotomic of the polar conjugate of X(2197)
= isogonal of the polar conjugate of X(26942)
= X(i)-Ceva conjugate of X(j) for these (i,j): {59, 23067}, {72, 201}, {26942, 2197}
= X(2632)-cross conjugate of X(520)
= X(i)-isoconjugate of X(j) for these (i,j): {4, 270}, {11, 24000}, {21, 8747}, {27, 1172}, {28, 29}, {58, 1896}, {60, 158}, {81, 8748}, {92, 2189}, {107, 3737}, {261, 1096}, {278, 2326}, {286, 2299}, {333, 5317}, {393, 2185}, {757, 1857}, {823, 7252}, {873, 6059}, {1098, 1118}, {1364, 24021}, {1396, 2322}, {2052, 2150}, {2170, 23582}, {3271, 23999}, {4560, 24019}, {4858, 23964}
= crosspoint of X(i) and X(j) for these (i,j): {59, 23067}, {72, 3682}, {1214, 28786}
= crosssum of X(28) and X(8747)
= barycentric product X(i) X(j) for these {i,j}: {3, 26942}, {12, 394}, {59, 15526}, {63, 201}, {65, 3998}, {69, 2197}, {71, 307}, {72, 1214}, {73, 306}, {77, 3949}, {181, 3926}, {219, 6356}, {222, 3695}, {226, 3682}, {228, 1231}, {255, 6358}, {278, 4158}, {312, 7138}, {321, 22341}, {326, 2171}, {345, 1425}, {348, 3690}, {349, 4055}, {520, 4552}, {525, 23067}, {594, 1804}, {756, 7183}, {1089, 7125}, {1252, 1367}, {1254, 3719}, {1259, 6354}, {1260, 20618}, {1262, 7068}, {1409, 20336}, {1439, 3694}, {1441, 3990}, {1500, 7055}, {1813, 4064}, {2149, 17879}, {2632, 4564}, {3265, 4559}, {3269, 4998}, {3964, 8736}, {4024, 6517}, {4131, 21859}, {4551, 24018}, {4574, 17094}, {7335, 28654}
= barycentric quotient X(i) / X(j) for these {i,j}: {12, 2052}, {37, 1896}, {42, 8748}, {48, 270}, {59, 23582}, {71, 29}, {73, 27}, {181, 393}, {184, 2189}, {201, 92}, {212, 2326}, {228, 1172}, {255, 2185}, {394, 261}, {520, 4560}, {577, 60}, {822, 3737}, {1214, 286}, {1259, 7058}, {1364, 26856}, {1367, 23989}, {1400, 8747}, {1402, 5317}, {1409, 28}, {1410, 1396}, {1425, 278}, {1500, 1857}, {1804, 1509}, {2149, 24000}, {2171, 158}, {2197, 4}, {2200, 2299}, {2289, 1098}, {2318, 2322}, {2632, 4858}, {2972, 26932}, {3269, 11}, {3682, 333}, {3690, 281}, {3695, 7017}, {3926, 18021}, {3949, 318}, {3990, 21}, {3998, 314}, {4055, 284}, {4158, 345}, {4551, 823}, {4552, 6528}, {4559, 107}, {4564, 23999}, {6056, 7054}, {6356, 331}, {6517, 4610}, {7068, 23978}, {7109, 6059}, {7125, 757}, {7138, 57}, {7183, 873}, {7335, 593}, {8736, 1093}, {20975, 8735}, {22061, 14006}, {22341, 81}, {23067, 648}, {24018, 18155}, {26942, 264}

2). Yes. Center of the circle:

a^2 (a^16 b^4-7 a^14 b^6+21 a^12 b^8-35 a^10 b^10+35 a^8 b^12-21 a^6 b^14+7 a^4 b^16-a^2 b^18-4 a^16 b^2 c^2+14 a^14 b^4 c^2-15 a^12 b^6 c^2+4 a^10 b^8 c^2-5 a^8 b^10 c^2+14 a^6 b^12 c^2-9 a^4 b^14 c^2+b^18 c^2+a^16 c^4+14 a^14 b^2 c^4-38 a^12 b^4 c^4+39 a^10 b^6 c^4-33 a^8 b^8 c^4+20 a^6 b^10 c^4-4 a^4 b^12 c^4+7 a^2 b^14 c^4-6 b^16 c^4-7 a^14 c^6-15 a^12 b^2 c^6+39 a^10 b^4 c^6+6 a^8 b^6 c^6-13 a^6 b^8 c^6-15 a^4 b^10 c^6-11 a^2 b^12 c^6+16 b^14 c^6+21 a^12 c^8+4 a^10 b^2 c^8-33 a^8 b^4 c^8-13 a^6 b^6 c^8+42 a^4 b^8 c^8+5 a^2 b^10 c^8-26 b^12 c^8-35 a^10 c^10-5 a^8 b^2 c^10+20 a^6 b^4 c^10-15 a^4 b^6 c^10+5 a^2 b^8 c^10+30 b^10 c^10+35 a^8 c^12+14 a^6 b^2 c^12-4 a^4 b^4 c^12-11 a^2 b^6 c^12-26 b^8 c^12-21 a^6 c^14-9 a^4 b^2 c^14+7 a^2 b^4 c^14+16 b^6 c^14+7 a^4 c^16-6 b^4 c^16-a^2 c^18+b^2 c^18) : :

= lies on these lines: {20,2979}, {185,5667}, {577,6759}, {3087,10110}


Best regards,
Peter Moses.