HYACINTHOS 29328

[Antreas P. Hatzipolakis]:

 

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

Denote:

 

(O') = the pedal circle of P ( = circumcircle of A'B'C')

(Oa) = the circle centered at A' with radius B'C'
(Ob) = the circle centered at B' with radius C'A'
(Oc) = the circle centered at C' with radius A'B'

Ra = the radical axis of (O'), (Oa)
Rb = the radical axis of (O'), (Ob)
Rc = the radical axis of (O'), (Oc)

A* = Ra /\ B'C'
B* = Rb /\ C'A'

C* = Rc /\ A'B'

[Ercole Suppa]:

For P = I:

1. A"B"C", A*B*C* are perspective at Feuerbach point X(11)
2. ABC,  A*B*C* are perspective at X(650)
3. A'B'C', A*B*C* are perspectve at X(3022)

 

[APH]:

 

For P = O:
In this case: B'C' = BC/2, C'A' = CA/2, A'B' = AB/2, therefore the circles (Oa), (Ob), (Oc) are the circles with diameters BC, CA, AB, resp.
So it is the problem Hyacinthos 29324  

 

Loci:

Which is the locus of P such that:

1. A"B"C", A*B*C* are perspective ?
2. ABC,  A*B*C* are perspective ?
3. A'B'C', A*B*C* are perspectve ?

I think that H lies on the loci too.

 

 

[Ercole Suppa]: 

 

Hi Antreas,

the point H lies only on loci 2,3. 

 

We have:

 

1. The locus of P such that:A"B"C", A*B*C* are perspective is an excentral circum-curve of order 9 very complicated, passing through I = X(1) 

 

 

2. The locus of P such that ABC,  A*B*C* are perspective is: 

 

{Apollonian circles}∪{K004 = Darboux cubic}

 

*** the isodynamic points X(15), X(16)  do not lie on the locus since Q1(X(15))=(0:0:0), Q1(X(16))=(0:0:0)

 

*** Pairs (P,Q2(P)) where Q2 = Q2(P) denotes the perspector: {1,650}, {3,523}, {4,15451} 

 

*** Some points:

 

Q2(X(20)) = X(520)X(647) ∩ X(525)X(14939)

 

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

 

= lies on these lines: {520,647}, {525,14939}, {6587,14398}

 

= barycentric product X(520)*X(1885)

 

= barycentric quotient X(1885)/X(6528)

 

= (6-9-13)  search numbers:  [-6.2813243628005331071, 7.5858283980078774561, 1.2880099122714792214]

 

 

Q2(X(40)) = X(650)X(663) ∩ X(3669)X(30574)

 

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

 

= lies on these lines: {650,663}, {3669,30574}, {9373,22091}

 

= barycentric product X(4041)*X(24556)

 

= barycentric quotient X(24556)/X(4625)

 

= trilinear quotient X(3709)/X(24556)

 

= (6-9-13)  search numbers:  [-31.3742169163530708542, 28.8423183240377520824, -1.8466865502634165597]

 

 

3. Τhe locus such that A'B'C', A*B*C* are perspectve is the entire plane

 

*** Some pairs (P, Q2(P)), where Q2 = Q2(P) denotes the perspector:

 

{1,3022},{3,115},{4,130},{6,115},{30,512},{32,115},{36,5532},{39,115},{50,115},{52,115},{58,115},{61,115},{62,115},{110,2682},{182,115},{187,115},{216,115},{284,115},{371,115},{372,115},{386,115},{389,115},{500,115},{512,512},{513,512},{514,512},{515,512},{516,512},{517,512},{518,512},{519,512},{520,512},{521,512},{522,512},{523,512},{524,512},{525,512},{526,512},{527,512},{528,512},{529,512},{530,512},{531,512},{532,512},{533,512},{534,512},{535,512},{536,512},{537,512},{538,512},{539,512},{540,512},{541,512},{542,512},{543,512},{544,512},{545,512},{566,115},{567,115},{568,115},{569,115},{570,115},{571,115},{572,115},{573,115},{574,115},{575,115},{576,115},{577,115},{578,115},{579,115},{580,115},{581,115},{582,115},{583,115},{584,115},{674,512},{680,512},{688,512},{690,512},{696,512},{698,512},{700,512},{702,512},{704,512},{706,512},{708,512},{710,512},{712,512},{714,512},{716,512},{718,512},{720,512},{722,512},{724,512},{726,512},{730,512},{732,512},{734,512},{736,512},{740,512},{742,512},{744,512},{746,512},{752,512},{754,512},{758,512},{760,512},{766,512},{768,512},{772,512},{776,512},{778,512},{780,512},{782,512},{784,512},{786,512},{788,512},{790,512},{792,512},{794,512},{796,512},{800,115},{802,512},{804,512},{806,512},{808,512},{812,512},{814,512},{816,512},{818,512},{824,512},{826,512},{830,512},{832,512},{834,512},{838,512},{888,512},{891,512},{900,512},{912,512},{916,512},{918,512},{924,512},{926,512},{928,512},{952,512},{970,115},{971,512},{991,115}

 

Best regards,

Ercole Suppa