[Antreas P. Hatzipolakis]:
Let ABC be a triangle.
Denote:
A', B', C' = the midpoints of AO, BO, CO, resp.
Oa, Ob, Oc = the circumcenters of OBC, OCA, OAB, resp.
A"B"C" = the orthic triangle of A'B'C'
1. OaObOc, A"B"C" are perspective.
2. The reflections of OaA", ObB", OcC" in BC, CA, AB, resp. are concurrent.
3. The reflections of OaA", ObB", OcC" in B'C', C'A', A'B', resp. are concurrent.
4. The reflections of OaA", ObB", OcC" in MbMc, McMa, MaMb, resp. are concurrent.
[Ercole Suppa]
Hi Antreas,
1. The perspector of OaObOc, A"B"C" is the point
Q1=X(6644)
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2. The reflections of OaA", ObB", OcC" in BC, CA, AB, resp. concur at point
Q2 = MIDPOINT OF X(159) AND X(1351)
= a^2 (a^10-3 a^8 b^2+2 a^6 b^4+2 a^4 b^6-3 a^2 b^8+b^10-3 a^8 c^2+4 a^2 b^6 c^2-b^8 c^2+2 a^6 c^4-2 a^2 b^4 c^4+2 a^4 c^6+4 a^2 b^2 c^6-3 a^2 c^8-b^2 c^8+c^10) : : (barys)
= (3 R^2 SB+3 R^2 SC-SB SW-SC SW)S^2 + 4 R^2 SB SC SW : : (barys)
= 3*X[182]-X[3357], 3*X[597]-X[6247], X[3098]-3*X[23042], 9*X[5050]-X[13093], 9*X[5085]-5*X[8567], 3*X[5102]+X[9924], 3*X[5476]-X[18381], X[5878]+3*X[11179], X[9934]+X[13248], 3*X[10168]-2*X[25563], 3*X[10250]-5*X[22234], 7*X[10541]-3*X[10606], 3*X[11216]-5*X[11482], X[14216]-3*X[23327], 2*X[18583]-X[23300], X[20299]-2*X[25555]
= lies on these lines: {3,1177}, {4,6}, {20,22151}, {23,154}, {24,18374}, {25,15135}, {26,206}, {64,7527}, {66,5576}, {141,3549}, {155,524}, {156,14984}, {157,30258}, {159,195}, {161,3060}, {182,3357}, {184,11470}, {378,15138}, {381,18432}, {389,19136}, {394,7493}, {542,8548}, {575,6000}, {576,2393}, {597,6247}, {599,7552}, {1350,7488}, {1352,10024}, {1598,19362}, {1614,2904}, {1619,11402}, {1853,5169}, {1971,13330}, {1974,19161}, {1994,7519}, {1995,15139}, {2777,25556}, {2888,11061}, {3098,23042}, {3172,28343}, {3564,15761}, {3589,15805}, {3818,9977}, {3843,32369}, {5050,13093}, {5085,8567}, {5102,9924}, {5198,11743}, {5476,18381}, {5622,6241}, {5878,11179}, {6145,7566}, {6293,7503}, {6403,20987}, {6800,10117}, {7387,9019}, {7529,16776}, {7540,9833}, {7556,17821}, {7564,19130}, {7691,19121}, {8537,14157}, {8540,26888}, {8541,26883}, {8547,15074}, {8721,22120}, {9715,32391}, {9934,13248}, {9971,10594}, {10168,25563}, {10250,22234}, {10510,12082}, {10535,19369}, {10541,10606}, {10601,23332}, {10752,11464}, {11216,11482}, {11799,18445}, {11819,21850}, {12294,21637}, {14216,23327}, {15066,17847}, {15068,16534}, {15647,19504}, {18583,23300}, {18911,32125}, {19164,22240}, {19347,32621}, {20299,25555}, {22802,32271}, {32046,32321}, {32251,32274}
= midpoint of X(i) and X(j) for these {i,j}: {159,1351}, {576,6759}, {9934,13248}, {11216,32063}
= reflection of X(i) in X(j) for these {i,j}: {64,15579}, {66,20300}, {9924,15580}, {15577,206}, {15581,6759}, {18382,5480}, {20299,25555}, {23300,18583}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {6,1181,8550}, {6,1498,8549}, {64,10249,15579}, {66,14561,20300}, {1350,19132,23041}, {5480,8550,12241}, {8549,19149,1498}
= (6-9-13) search numbers: [-3.7709681981125136147, -4.2394987650923212154, 8.3161489491771462283]
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3. The reflections of OaA", ObB", OcC" in B'C', C'A', A'B', resp. concur at point
Q3 = X(182)
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4. The reflections of OaA", ObB", OcC" in MbMc, McMa, MaMb, resp concur at point
Q4 = MIDPOINT OF X(66) AND X(1352)
= a^12-2 a^10 b^2+a^8 b^4-a^4 b^8+2 a^2 b^10-b^12-2 a^10 c^2+2 a^8 b^2 c^2-2 a^6 b^4 c^2+2 b^10 c^2+a^8 c^4-2 a^6 b^2 c^4+2 a^4 b^4 c^4-2 a^2 b^6 c^4+b^8 c^4-2 a^2 b^4 c^6-4 b^6 c^6-a^4 c^8+b^4 c^8+2 a^2 c^10+2 b^2 c^10-c^12 : : (barys)
= (3 R^2 SB+3 R^2 SC-4 R^2 SW-SB SW-SC SW+SW^2)S^2 -2 R^2 SB SC SW+SB SC SW^2 : : (barys)
= X[206]-2*X[24206], 5*X[1656]-3*X[19153], 5*X[3763]-3*X[23041], 3*X[3818]-X[22802], X[6759]-3*X[11178], X[11477]-3*X[23049], 3*X[14810]-2*X[32903], 5*X[17821]-9*X[21358]
= lies on these lines: {2,15139}, {3,66}, {4,67}, {6,70}, {68,524}, {69,1225}, {154,7495}, {182,6689}, {195,15141}, {206,24206}, {343,26283}, {394,858}, {511,9927}, {542,1147}, {575,18952}, {576,12585}, {631,32337}, {924,18312}, {1216,2393}, {1350,12225}, {1656,19153}, {1899,5094}, {2854,15133}, {3410,16063}, {3448,9716}, {3549,19127}, {3564,13371}, {3763,23041}, {3818,22802}, {3827,5694}, {5480,7507}, {5596,7558}, {5622,23294}, {5921,28419}, {5925,6240}, {6000,18431}, {6293,7544}, {6640,15462}, {6759,11178}, {7505,18374}, {7528,16776}, {7545,32262}, {7729,14982}, {9019,14790}, {9630,12589}, {10249,11457}, {10295,10606}, {10516,13160}, {11470,32285}, {11477,23049},{14003,20021}, {14810,32903}, {15073,25739}, {15106,23315}, {17821,21358}, {18475,18580}, {18909,32184}, {18919,22533}, {21284,31383}, {31670,31724}
= midpoint of X(i) and X(j) for these {i,j}: {66,1352}, {8549,15069}, {15141,32306}
= reflection of X(i) in X(j) for these {i,j}: {6,20300}, {182,6697}, {206,24206}, {575,32767}, {9833,15582}, {15577,141}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1853,15069,8549}
= (6-9-13) search numbers: [4.8767116813823228036, 3.6032854233383545696, -1.1047077410417076621]
Best regards
Ercole Suppa
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