HYACINTHOS 28619

[Antreas P. Hatzipolakis]:

 

 

Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.

Denote:

D = the Poncelet point of ABCP

A"B"C" = the medial triangle of A'B'C'

La =: DA", Lb =: DB", Lc =: DC"

L1, L2, L3 = the reflections of La, Lb, Lc in PA', PB', PC', resp. (or in BC, CA, AB, resp.)

The reflections of L1, L2, L3 in BC, CA, AB, resp. (or in PA', PB', PC', resp.) are concurrent.

Point of concurrence in terms of P?


[Peter Moses]:

Hi Antreas,

P = (p:q:r) -->

a^2 (-5 a^2 b^2 c^4 p^3 q^2+5 b^4 c^4 p^3 q^2+3 b^2 c^6 p^3 q^2-5 a^2 b^2 c^4 p^2 q^3+5 b^4 c^4 p^2 q^3+2 a^2 c^6 p^2 q^3-3 b^2 c^6 p^2 q^3-2 c^8 p^2 q^3+8 a^4 b^2 c^2 p^3 q r-11 a^2 b^4 c^2 p^3 q r+3 b^6 c^2 p^3 q r-11 a^2 b^2 c^4 p^3 q r+10 b^4 c^4 p^3 q r+3 b^2 c^6 p^3 q r+2 a^6 c^2 p^2 q^2 r+6 a^4 b^2 c^2 p^2 q^2 r-8 a^2 b^4 c^2 p^2 q^2 r-6 a^4 c^4 p^2 q^2 r-3 a^2 b^2 c^4 p^2 q^2 r+5 b^4 c^4 p^2 q^2 r+6 a^2 c^6 p^2 q^2 r-3 b^2 c^6 p^2 q^2 r-2 c^8 p^2 q^2 r+2 a^6 c^2 p q^3 r-2 a^4 b^2 c^2 p q^3 r+3 a^2 b^4 c^2 p q^3 r-3 b^6 c^2 p q^3 r-2 a^4 c^4 p q^3 r-5 a^2 b^2 c^4 p q^3 r+4 b^4 c^4 p q^3 r+2 a^2 c^6 p q^3 r+b^2 c^6 p q^3 r-2 c^8 p q^3 r-5 a^2 b^4 c^2 p^3 r^2+3 b^6 c^2 p^3 r^2+5 b^4 c^4 p^3 r^2+2 a^6 b^2 p^2 q r^2-6 a^4 b^4 p^2 q r^2+6 a^2 b^6 p^2 q r^2-2 b^8 p^2 q r^2+6 a^4 b^2 c^2 p^2 q r^2-3 a^2 b^4 c^2 p^2 q r^2-3 b^6 c^2 p^2 q r^2-8 a^2 b^2 c^4 p^2 q r^2+5 b^4 c^4 p^2 q r^2+2 a^6 b^2 p q^2 r^2-6 a^4 b^4 p q^2 r^2+6 a^2 b^6 p q^2 r^2-2 b^8 p q^2 r^2+2 a^6 c^2 p q^2 r^2+12 a^4 b^2 c^2 p q^2 r^2-6 a^2 b^4 c^2 p q^2 r^2-2 b^6 c^2 p q^2 r^2-6 a^4 c^4 p q^2 r^2-6 a^2 b^2 c^4 p q^2 r^2+8 b^4 c^4 p q^2 r^2+6 a^2 c^6 p q^2 r^2-2 b^2 c^6 p q^2 r^2-2 c^8 p q^2 r^2+2 a^6 c^2 q^3 r^2+2 a^4 b^2 c^2 q^3 r^2-2 a^4 c^4 q^3 r^2+2 a^2 b^6 p^2 r^3-2 b^8 p^2 r^3-5 a^2 b^4 c^2 p^2 r^3-3 b^6 c^2 p^2 r^3+5 b^4 c^4 p^2 r^3+2 a^6 b^2 p q r^3-2 a^4 b^4 p q r^3+2 a^2 b^6 p q r^3-2 b^8 p q r^3-2 a^4 b^2 c^2 p q r^3-5 a^2 b^4 c^2 p q r^3+b^6 c^2 p q r^3+3 a^2 b^2 c^4 p q r^3+4 b^4 c^4 p q r^3-3 b^2 c^6 p q r^3+2 a^6 b^2 q^2 r^3-2 a^4 b^4 q^2 r^3+2 a^4 b^2 c^2 q^2 r^3) : :

P = X(1) -> 

 

= REFLECTION OF X(11) IN X(17660)

 

a (2 a^4 b-4 a^3 b^2+4 a b^4-2 b^5+2 a^4 c+2 a^2 b^2 c-5 a b^3 c+b^4 c-4 a^3 c^2+2 a^2 b c^2+2 a b^2 c^2+b^3 c^2-5 a b c^3+b^2 c^3+4 a c^4+b c^4-2 c^5) : : 

= 5 X[11] - 6 X[354], 5 X[100] - 3 X[4661], 9 X[354] - 10 X[5083], 3 X[11] - 4 X[5083], 5 X[1317] - 4 X[9957], 2 X[6797] - 3 X[11570], 3 X[5903] - 5 X[11571], 6 X[9957] - 5 X[12758], 3 X[1317] - 2 X[12758], 3 X[354] - 5 X[17660], 2 X[5083] - 3 X[17660], 7 X[11] - 8 X[18240], 7 X[5083] - 6 X[18240], 7 X[17660] - 4 X[18240].

 

= lies on these lines: {11,118}, {55,13243}, {80,10404}, {100,4661}, {518,6154}, {952,5903}, {1317,2771}, {2800,12680}, {3614,12005}, {3874,12690}, {5432,13226}, {5904,9945}, {6797,11570}, {7672,11246}, {9803,12763}, {9809,13274}, {9946,14872}, {9964,10950}, {10864,13253}, {12675,12691}, {12750,16128}

= reflection of X(i) in X(j) for these {i,j}: {11, 17660}, {5904, 9945}, {12690, 3874}, {12691, 12675}, {14872, 9946}


P = X(2) -> 

 

= X(99)X(11593)∩X(115)X(373)

 

a^2 (2 a^6 b^2-4 a^4 b^4+4 a^2 b^6-2 b^8+2 a^6 c^2+8 a^4 b^2 c^2-10 a^2 b^4 c^2-b^6 c^2-4 a^4 c^4-10 a^2 b^2 c^4+14 b^4 c^4+4 a^2 c^6-b^2 c^6-2 c^8) : : 

= 5 X[115] - 6 X[373], 5 X[2482] - 4 X[3819], 16 X[10219] - 15 X[14971], 2 X[12162] - 5 X[14981], 5 X[14928] - 2 X[17710].

 

= lies on these lines: {99,11593}, {115,373}, {543,3060}, {2482,3819}, {2936,8780}, {10219,14971}, {12162,14981}, {14928,17710}

Best regards,
Peter Moses.