HYACINTHOS 28904

[Seiichi Kirikami]:

Dear friends,

Let ABC be a triangle and P its X(195) ( on Napoleon - Feuerbach cubic K005 )

Denote:

Pa, Pb, Pc = the orthogonal projections of P on BC, CA, AB respectively.

Na, Nb, Nc = the nine-point centers of APbPc, BPcPa, CPaPb respectively.

la, lb, lc = the Euler lines of APbPc, BPcPa, CPaPb respectively.

ha, hb, hc = the Hatzipolakis axes of APbPc, BPcPa, CPaPb respectively.

la, lb and lc concur in a point Q.

ha, hb and hc concur in a point R.

Q, R, Na, Nb and Nc are concyclic with the diameter QR and the center S ( = X(1493))

(6-9-13) search numbers of Q = (-12.9811249473..,5.4675605072..,5.8467187219..).

(6-9-13) search numbers of R = (-12.3645151043..,20.900601246..,-5.1222832681..).

Hatzipolakis axis is the line perpendicular to Euler line through the nine-point center.

X(195) is a triangle center in Encyclopedia of Triangle Centers and K005 is a cubic in Catalogue of Triangle Cubics.

Best regards, Seiichi 


[Peter Moses]:

Hi Antreas,

Point Q = MIDPOINT OF X(6343) AND X(11671)

 

= 2 a^22-15 a^20 b^2+52 a^18 b^4-115 a^16 b^6+188 a^14 b^8-238 a^12 b^10+224 a^10 b^12-142 a^8 b^14+50 a^6 b^16-3 a^4 b^18-4 a^2 b^20+b^22-15 a^20 c^2+82 a^18 b^2 c^2-192 a^16 b^4 c^2+249 a^14 b^6 c^2-165 a^12 b^8 c^2-31 a^10 b^10 c^2+173 a^8 b^12 c^2-133 a^6 b^14 c^2+16 a^4 b^16 c^2+25 a^2 b^18 c^2-9 b^20 c^2+52 a^18 c^4-192 a^16 b^2 c^4+284 a^14 b^4 c^4-218 a^12 b^6 c^4+108 a^10 b^8 c^4-121 a^8 b^10 c^4+176 a^6 b^12 c^4-72 a^4 b^14 c^4-52 a^2 b^16 c^4+35 b^18 c^4-115 a^16 c^6+249 a^14 b^2 c^6-218 a^12 b^4 c^6+100 a^10 b^6 c^6+9 a^8 b^8 c^6-112 a^6 b^10 c^6+150 a^4 b^12 c^6+12 a^2 b^14 c^6-75 b^16 c^6+188 a^14 c^8-165 a^12 b^2 c^8+108 a^10 b^4 c^8+9 a^8 b^6 c^8+38 a^6 b^8 c^8-91 a^4 b^10 c^8+120 a^2 b^12 c^8+90 b^14 c^8-238 a^12 c^10-31 a^10 b^2 c^10-121 a^8 b^4 c^10-112 a^6 b^6 c^10-91 a^4 b^8 c^10-202 a^2 b^10 c^10-42 b^12 c^10+224 a^10 c^12+173 a^8 b^2 c^12+176 a^6 b^4 c^12+150 a^4 b^6 c^12+120 a^2 b^8 c^12-42 b^10 c^12-142 a^8 c^14-133 a^6 b^2 c^14-72 a^4 b^4 c^14+12 a^2 b^6 c^14+90 b^8 c^14+50 a^6 c^16+16 a^4 b^2 c^16-52 a^2 b^4 c^16-75 b^6 c^16-3 a^4 c^18+25 a^2 b^2 c^18+35 b^4 c^18-4 a^2 c^20-9 b^2 c^20+c^22 : : 

= X[2888] - 3 X[25147]

 

= lies on the curve Q093 and these  lines: {4,195}, {30,27246}, {54,16766}, {1154,12026}, {1157,27196}, {2888,25147}, {3459,21230}, {6343,11671}, {6592,8254}, {7691,16768}, {14072,16762}, {22051,25150}

= midpoint of X(6343) and X(11671)
= reflection of X(6592) in X(8254)


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Point R = REFLECTION OF X(12026) IN X(54)

 

= (2 a^10-7 a^8 b^2+10 a^6 b^4-8 a^4 b^6+4 a^2 b^8-b^10-7 a^8 c^2+10 a^6 b^2 c^2-a^4 b^4 c^2-5 a^2 b^6 c^2+3 b^8 c^2+10 a^6 c^4-a^4 b^2 c^4+2 a^2 b^4 c^4-2 b^6 c^4-8 a^4 c^6-5 a^2 b^2 c^6-2 b^4 c^6+4 a^2 c^8+3 b^2 c^8-c^10) (3 a^12-12 a^10 b^2+19 a^8 b^4-16 a^6 b^6+9 a^4 b^8-4 a^2 b^10+b^12-12 a^10 c^2+20 a^8 b^2 c^2-4 a^6 b^4 c^2-10 a^4 b^6 c^2+12 a^2 b^8 c^2-6 b^10 c^2+19 a^8 c^4-4 a^6 b^2 c^4+5 a^4 b^4 c^4-8 a^2 b^6 c^4+15 b^8 c^4-16 a^6 c^6-10 a^4 b^2 c^6-8 a^2 b^4 c^6-20 b^6 c^6+9 a^4 c^8+12 a^2 b^2 c^8+15 b^4 c^8-4 a^2 c^10-6 b^2 c^10+c^12) : :

= lies on these lines: {5,49}, {195,6343}, {5898,14367}, {6150,6592}, {20414,22051}

= reflection of X(12026) in X(54)


Best regards,
Peter Moses.