HYACINTHOS 29153

[Antreas P. Hatzipolakis]:

 

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

Denote:

(Na), (Nb), (Nc) = the NPCs of PBC, PCA, PAB, resp.

D = the Poncelet point of ABCP.

PA' intersects (Na) again at A"
PB' intersects (Nb) again at B"
PC' intersects (Nc) again at C"

Which is the locus of P such that the reflections of DA", DB", DC" in AP, BP, CP, resp. are concurrent?

I lies on the locus (the reflections are parallels to OI line)
G lies on the locus.

 

 [César Lozada]:

 

Locus = {Linf} ∪ {circumcircle} ∪ {Q003 through ETC’s 1, 2, 4, 13, 14, 357, 1113, 1114, 1134, 1136, 1156 } ∪ {q4: circum-quartic through X(4)}

q4: ∑ [ y*z*((SB+SC)^2*y*z+2*(SA^2-S^2)*x^2) ] = 0 (barys)

 

Let Q(P) be the point of concurrence.

If P lies on the circumcircle then PQ = (-1/2)*PH, ie, Q(P) lies on the circle with center X(550) and radius=(3/2)*R.  

 

ETC pairs (P,Q(P)) for P on the circumcircle: (74,10990), (98,10991), (99,10992), (100,10993), (110,30714), (112,14900)

 

Some others:

Q( X(1) ) =X(517)

Q( X(2) ) =X(187)

 

Q( X(13) ) = MIDPOINT OF X(6778) AND X(23005)

= sqrt(3)*(2*a^4-(b^2+c^2)*a^2+(b^2-c^2)^2)*(-a^2+b^2+c^2)+2*S*(2*a^4-(b^2+c^2)*a^2-3*(b^2-c^2)^2) : : (barys)

= 3*X(13)-X(16), 5*X(13)-X(6779), 2*X(16)-3*X(6108), 5*X(16)-3*X(6779), 3*X(115)-2*X(11543), 2*X(5321)-3*X(31710), 3*X(5459)-2*X(6672), 9*X(5470)-5*X(16961), 3*X(5470)-X(22998), 5*X(6108)-2*X(6779), 3*X(6778)+X(19107), 3*X(6782)-4*X(11543), 3*X(11603)-X(22856), 5*X(16961)-3*X(22998), X(19107)-3*X(23005), 3*X(22511)-X(25235)

= lies on these lines: {2, 13}, {6, 20429}, {30, 6783}, {115, 6782}, {187, 11542}, {396, 32907}, {397, 25555}, {511, 5318}, {524, 31695}, {533, 23004}, {542, 5321}, {925, 2379}, {3564, 5107}, {5470, 16961}, {5472, 14136}, {5873, 7748}, {6772, 11485}, {6778, 10722}, {7685, 16808}, {8586, 20428}, {11603, 22856}, {13103, 22513}, {13349, 20415}, {14139, 23303}, {14447, 27551}, {20412, 31687}, {22511, 25235}

= midpoint of X(6778) and X(23005)

= reflection of X(i) in X(j) for these (i,j): (187, 11542), (6108, 13), (6782, 115), (13349, 20415)

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (16, 16966, 6672), (622, 5335, 16)

≈ [ 3.5060234756263950, 2.3285920314464330, 0.4103976252323584 ]

 

Q( X(14) ) = MIDPOINT OF X(6777) AND X(23004)

= sqrt(3)*(2*a^4-(b^2+c^2)*a^2+(b^2-c^2)^2)*(-a^2+b^2+c^2)-2*S*(2*a^4-(b^2+c^2)*a^2-3*(b^2-c^2)^2) : : (barys)

= 3*X(14)-X(15), 5*X(14)-X(6780), 2*X(15)-3*X(6109), 5*X(15)-3*X(6780), 3*X(115)-2*X(11542), 2*X(5318)-3*X(31709), 3*X(5460)-2*X(6671), 9*X(5469)-5*X(16960), 3*X(5469)-X(22997), 5*X(6109)-2*X(6780), 3*X(6777)+X(19106), 3*X(6783)-4*X(11542), 3*X(11602)-X(22900), 5*X(16960)-3*X(22997), X(19106)-3*X(23004), 3*X(22510)-X(25236)

= lies on these lines: {2, 14}, {6, 20428}, {30, 6782}, {115, 6783}, {187, 11543}, {395, 32909}, {398, 25555}, {511, 5321}, {524, 31696}, {532, 23005}, {542, 5318}, {925, 2378}, {3564, 5107}, {5469, 16960}, {5471, 14137}, {5872, 7748}, {6775, 11486}, {6777, 10722}, {7684, 16809}, {8586, 20429}, {11602, 22900}, {13102, 22512}, {13350, 20416}, {14138, 23302}, {14446, 27550}, {20411, 31688}, {22510, 25236}

= midpoint of X(6777) and X(23004)

= reflection of X(i) in X(j) for these (i,j): (187, 11543), (6109, 14), (6783, 115), (13350, 20416)

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (15, 16967, 6671), (621, 5334, 15)

≈ [ 5.0089256476927410, 8.5018257131261500, -4.5570266953457610 ]

 

Q( X(1156) ) = REFLECTION OF X(5902) IN X(12019)

= a*( 2*(b+c)*a^4-4*(b^2+c^2)*a^3+2*(b+c)*b*c*a^2+(4*b^2+7*b*c+4*c^2)*(b-c)^2*a-(b^2-c^2)*(b-c)*(b+2*c)*(2*b+c)) :: (barys)

= 3*X(11)-2*X(354), 7*X(11)-4*X(5083), 5*X(11)-2*X(17660), 11*X(11)-8*X(18240), 4*X(11)-X(27778), 7*X(354)-6*X(5083), 5*X(354)-3*X(17660), 11*X(354)-12*X(18240), 8*X(354)-3*X(27778), 4*X(3740)-3*X(6174), X(4430)-3*X(10707), 10*X(5083)-7*X(17660), 11*X(5083)-14*X(18240), 16*X(5083)-7*X(27778), 11*X(17660)-20*X(18240), 8*X(17660)-5*X(27778)

= lies on these lines: {11, 118}, {55, 1156}, {80, 1836}, {100, 3715}, {528, 3681}, {952, 3058}, {1898, 18908}, {2802, 4525}, {3740, 6174}, {4430, 10707}, {4995, 15064}, {5777, 10543}, {5851, 11246}, {5902, 12019}, {6326, 30326}, {9580, 9897}, {10176, 10609}

= reflection of X(i) in X(j) for these (i,j): (5902, 12019), (10609, 10176)

= [ 5.3225098087895330, 7.3765166558148830, -3.9226977300210930 ]

 

Q( X(101)  ) = REFLECTION OF X(118) IN X(101)

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

= X(4)-3*X(101), 2*X(4)-3*X(118), 5*X(4)-3*X(10725), 5*X(101)-X(10725), 3*X(103)-5*X(3522), 3*X(116)-4*X(140), 5*X(118)-2*X(10725), 3*X(150)-7*X(3523), 3*X(152)+X(5059), 3*X(381)-4*X(20401), 5*X(631)-3*X(10708), X(962)-3*X(15735), 5*X(1656)-6*X(6710), 5*X(1656)-3*X(10739), X(3146)-3*X(10710), 5*X(3522)+3*X(20096), 7*X(3523)-6*X(6712), 17*X(3533)-15*X(31273), X(5073)-3*X(10741), 5*X(11522)-6*X(11728)

= lies on these lines: {3, 544}, {4, 101}, {72, 2801}, {103, 3522}, {116, 140}, {150, 3523}, {152, 5059}, {381, 20401}, {516, 6603}, {550, 2808}, {631, 10708}, {952, 31852}, {962, 15735}, {1656, 6710}, {2772, 10990}, {2774, 30714}, {2784, 3704}, {2786, 10992}, {2809, 5882}, {2810, 8550}, {3146, 10710}, {3533, 31273}, {3887, 10993}, {4292, 15730}, {4294, 4845}, {5073, 10741}, {9518, 14900}, {11522, 11728}, {11712, 13464}

= midpoint of X(103) and X(20096)

= reflection of X(i) in X(j) for these (i,j): (118, 101), (150, 6712), (10739, 6710)

= [ 4.8190698099908630, 1.5677877589333180, 0.3310868903424531 ]

 

 

Q( X(103)  ) = REFLECTION OF X(116) IN X(103)

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

= X(4)-3*X(103), 2*X(4)-3*X(116), 5*X(4)-3*X(10727), 3*X(101)-5*X(3522), 5*X(103)-X(10727), 5*X(116)-2*X(10727), 3*X(118)-4*X(140), 3*X(150)+X(5059), 3*X(152)-7*X(3523), 5*X(631)-3*X(10710), 5*X(631)-4*X(20401), 5*X(1656)-6*X(6712), 5*X(1656)-3*X(10741), X(3146)-3*X(10708), 7*X(3523)-6*X(6710), 13*X(5068)-15*X(31273), X(5073)-3*X(10739), 3*X(10710)-4*X(20401), 5*X(11522)-6*X(11726), 3*X(11714)-2*X(13464)

= lies on these lines: {4, 103}, {20, 544}, {101, 3522}, {118, 140}, {150, 5059}, {152, 3523}, {550, 2808}, {631, 10710}, {1656, 6712}, {2772, 30714}, {2774, 10990}, {2784, 10992}, {2786, 10991}, {2801, 10993}, {2809, 5493}, {2825, 14900}, {3146, 10708}, {5068, 31273}, {5073, 10739}, {11522, 11726}, {11714, 13464}

= reflection of X(i) in X(j) for these (i,j): (116, 103), (152, 6710), (10741, 6712)

= {X(631), X(10710)}-harmonic conjugate of X(20401)

= [ 21.2046382835247100, 22.6482292897462600, -21.8256350033898100 ]

 

César Lozada