ADGEOM 4921 * ADGEOM 4925 * ADGEOM 4927 * ADGEOM 4941

#4921

Dear geometers,

 

Let ABC be a triangle then 

 

Centroid G, symmedian point S, circumcenter O and Parry point P are concyclic.

 

Is this result known before?

 

Best regards,

Tran Quang Hung.

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#4925

Center = X(9175)

Square-radius = 4*R^2* S^2* (3*S^2-SW^2)* (9*R^2-2*SW)*(-SW^3+3*(9*R^2-SW)*S^2)/K^2, where K=3*SW*∏(|SB-SC|)

Through ETC’s: {2, 3, 6, 111, 691, 5653, 9173, 9174, 9178, 11579, 11632, 11637, 11638, 14174, 14180, 14699, 14700, 15546, 15744}

 

Some others:

Antipode of X(2) =

P01 = X(3)X(523) ∩ X(98)X(111)

= (3*a^10-4*(b^2+c^2)*a^8-(2*b^4-7*b^2*c^2+2*c^4)*a^6+(b^2+c^2)*(4*b^4-5*b^2*c^2+4*c^4)*a^4-(b^8+c^8-b^2*c^2*(7*b^4-16*b^2*c^2+7*c^4))*a^2-(b^4-c^4)*(b^2-c^2)*b^2*c^2)*(b^2-c^2) : : (barys)

= 3*X(3545)-2*X(18309)

= on lines: {2, 9137}, {3, 523}, {4, 2489}, {6, 1499}, {30, 9178}, {98, 111}, {182, 5652}, {512, 11179}, {542, 5653}, {690, 11579}, {691, 2407}, {804, 11632}, {1352, 11182}, {2492, 19912}, {3545, 18309}, {5996, 9744}, {6088, 14666}, {7709, 8704}, {8371, 15928}, {9171, 20423}, {12106, 21006}

= reflection of X(i) in X(j) for these (i,j): (2, 9175), (1352, 11182)

= [ 3.6274797921220480, 9.0737346583668690, -4.3153732625567060 ]

 

Antipode of X(6) =

P02 = X(3)X(512) ∩ X(74)X(111)

= a^2*(a^8+3*(b^2+c^2)*a^6-(11*b^4+5*b^2*c^2+11*c^4)*a^4+(b^2+c^2)*(9*b^4-2*b^2*c^2+9*c^4)*a^2-2*b^8+3*b^6*c^2-14*b^4*c^4+3*b^2*c^6-2*c^8)*(b^2-c^2) : : (barys)

= 3*X(5054)-2*X(11183), 3*X(10516)-2*X(18309)

= on lines: {2, 1499}, {3, 512}, {6, 9175}, {74, 111}, {381, 11182}, {511, 9178}, {525, 16220}, {526, 11579}, {549, 5652}, {690, 11632}, {691, 2421}, {804, 19905}, {924, 10249}, {1351, 9171}, {5054, 11183}, {5653, 5663}, {6785, 14700}, {9126, 9135}, {9137, 15066}, {10516, 18309}

= reflection of X(i) in X(j) for these (i,j): (6, 9175), (381, 11182), (1351, 9171)

= [ 5.2639400895450970, 9.3372844440941770, -5.2531201745631710 ]

 

César Lozada

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#4927

TCCT  Table 8.1  Central circles that pass through the Parry point

 

Angel Montesdeoca

 

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#4941

 

Dear César 

in the lists reflection of X(i) in X(j) op points P01 and P02 I have find other pairs:

 

Antipode of X(2) = P01 

 = reflection of X(i) in X(j) for these (i,j): {2, 9175}, {1352, 11182}, {5652, 182}, {19912, 2492}, {20423, 9171}

 

 

Antipode of X(6) = P02 

 = reflection of X(i) in X(j) for these (i,j):  {6, 9175}, {381, 11182}, {1351, 9171}, {5652, 549}, {9135, 9126}

 

 

Best regards

Ercole Suppa