HYACINTHOS 28981

[Kadir Altintas]:

 

 

Let ABC be a triangle with O, H the  circumcenter, orthocenter, resp. 

DEF is the orthic triangle of ABC

Circle passing through E,F,A' intersects the circumcircle at A''. Define B'', C'' cyclically

 

(1) Prove that AA', BB'', CC'' concur at a point X.

 

(2) What is the locus for P such that DEF is cevian triangle of P and property above holds?

 

Romantics of Geometry, problem 2952

 

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[Ercole Suppa]

 

The locus of points P such that AA', BB'', CC'' are concurrent = {q9: circum curve of order 9, through X(4) and X(20)}

 

q9: ∑ 2 c^2 (2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4) x^5 y^3 z+6 (a-b) (a+b) c^2 (a^2+b^2-3 c^2) x^4 y^4 z+2 c^2 (a^4+a^2 b^2-2 b^4-2 a^2 c^2+b^2 c^2+c^4) x^3 y^5 z-4 (a-b) (a+b) (a-c) (b-c) (a+c) (b+c) x^5 y^2 z^2-(a^6+9 a^4 b^2-9 a^2 b^4-b^6-23 a^4 c^2+6 a^2 b^2 c^2+17 b^4 c^2+19 a^2 c^4-27 b^2 c^4+3 c^6) x^4 y^3 z^2-(a^6+9 a^4 b^2-9 a^2 b^4-b^6-17 a^4 c^2-6 a^2 b^2 c^2+23 b^4 c^2+27 a^2 c^4-19 b^2 c^4-3 c^6) x^3 y^4 z^2-36 a^2 (a-b) b^2 (a+b) x^3 y^3 z^3-2 a^4 (a^2+b^2-c^2) y^5 z^4+2 a^4 (a^2-b^2+c^2) y^4 z^5 = 0 (barys)

 

Let Q(X(i)) be the perspector of ABC and A''B''C''. We have:

 

Q(X(4)) = X(393)

 

 

Q(X(20)) = X(64)X(28783) ∩ X(154)X(1033)

 

= a^2 (a^12+6 a^10 b^2-29 a^8 b^4+36 a^6 b^6-9 a^4 b^8-10 a^2 b^10+5 b^12-6 a^10 c^2+14 a^8 b^2 c^2+4 a^6 b^4 c^2-36 a^4 b^6 c^2+34 a^2 b^8 c^2-10 b^10 c^2+15 a^8 c^4-20 a^6 b^2 c^4+50 a^4 b^4 c^4-36 a^2 b^6 c^4-9 b^8 c^4-20 a^6 c^6-20 a^4 b^2 c^6+4 a^2 b^4 c^6+36 b^6 c^6+15 a^4 c^8+14 a^2 b^2 c^8-29 b^4 c^8-6 a^2 c^10+6 b^2 c^10+c^12) (a^12-6 a^10 b^2+15 a^8 b^4-20 a^6 b^6+15 a^4 b^8-6 a^2 b^10+b^12+6 a^10 c^2+14 a^8 b^2 c^2-20 a^6 b^4 c^2-20 a^4 b^6 c^2+14 a^2 b^8 c^2+6 b^10 c^2-29 a^8 c^4+4 a^6 b^2 c^4+50 a^4 b^4 c^4+4 a^2 b^6 c^4-29 b^8 c^4+36 a^6 c^6-36 a^4 b^2 c^6-36 a^2 b^4 c^6+36 b^6 c^6-9 a^4 c^8+34 a^2 b^2 c^8-9 b^4 c^8-10 a^2 c^10-10 b^2 c^10+5 c^12) : : (barys)

 

= (8 R^2-SB-SC-2 SW)S^4 + (512 R^6-128 R^4 SB-128 R^4 SC-256 R^4 SW+48 R^2 SB SW+48 R^2 SC SW+2 SB SC SW+32 R^2 SW^2-4 SB SW^2-4 SC SW^2)S^2 + 4096 R^8 SB+4096 R^8 SC+1024 R^6 SB SC-3072 R^6 SB SW-3072 R^6 SC SW-384 R^4 SB SC SW+768 R^4 SB SW^2+768 R^4 SC SW^2+32 R^2 SB SC SW^2-64 R^2 SB SW^3-64 R^2 SC SW^3 : : (barys)

 

= lies on thee lines: {64,28783}, {154,1033}, {1073,14481}, {1249,3349}, {1498,3348}, {3197,8803}

= barycentric product of X(i) and X(j) for these {i,j}: {4,3348}, {64,14365}, {253,28781}, {3346,3349}

= barycentric quotient of X(i) and X(j) for these {i,j}: {25,3183}, {64,14362}, {154,2060}, {1042,8812}, {3348,69}, {3349,6527},{14365,14615}, {28781,20}

= trilinear product of X(i) and X(j) for these {i,j}: {19,3348}, {2155,14365}, {2184,28781}

= trilinear quotient of X(i) and X(j) for these {i,j}: {1427,8812}, {14365,18750}

 

= (6-9-13) search numbers [0.119372859269926633,0.219812364735888389,3.43339152511956716]

 

 

Best regards

Ercole Suppa