HYACINTHOS 28514

 

[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.

Da, Db, Dc = the antipodes of D in (Na), (Nb), (Nc), resp.

Which is the locus of P such that A'B'C', DaDbDc  are perspective?  

 

[Angel Montesdeoca] 

****The locus of P such that A'B'C', DaDbDc  are perspective is the entire plane.

Notation:
M a point
M* is the isogonal conjugate of M
cM is the  complement of M
aM is the anticomplement of M

The perspector Q of A'B'C' and DaDbDc is:

P=(x:y:z) --> Q = (c^2 x y^2-2 a^2 x y z+b^2 x y z+c^2 x y z-a^2 y^2 z+b^2 x z^2-a^2 y z^2) (-a^2 c^2 x^3 y+b^2 c^2 x^3 y+c^4 x^3 y-a^2 c^2 x^2 y^2+b^2 c^2 x^2 y^2-c^4 x^2 y^2-a^2 b^2 x^3 z+b^4 x^3 z+b^2 c^2 x^3 z-2 a^4 x^2 y z+2 a^2 b^2 x^2 y z+2 a^2 c^2 x^2 y z-2 a^4 x y^2 z+3 a^2 b^2 x y^2 z-b^4 x y^2 z-a^2 c^2 x y^2 z+2 b^2 c^2 x y^2 z-c^4 x y^2 z-a^2 b^2 x^2 z^2-b^4 x^2 z^2+b^2 c^2 x^2 z^2-2 a^4 x y z^2-a^2 b^2 x y z^2-b^4 x y z^2+3 a^2 c^2 x y z^2+2 b^2 c^2 x y z^2-c^4 x y z^2-2 a^4 y^2 z^2) : ... : ...

P --> Q = midpoint of  (reflection of  aP* in P) and (Collings transform of cP)

If P lies on circumcirle,  Q = P*

If P lies on the line at infinity, Q = P

Other pairs {P=X(i), Q=X(j)}, for {i,j}: 
{1,1317}, {2,18800}, {3,113}, {5,14071}, {6,15303}, {7,18801}, {8,18802}, {13,9117}, {14,9115}, {15,18803}, {16,18804}, {54,11702}, {671,22329}, {1263,24147}.

Some others:

**   P = X (9)  ---> Q9  =  MIDPOINT OF X(100) AND X(12848)

(2 a^2 - (b - c)^2 - a (b + c)) (2 a^6 - 5 a^5 (b + c) - a (b - c)^4 (b + c) + (b - c)^4 (b + c)^2 + a^4 (b^2 + 14 b c + c^2) + 2 a^3 (3 b^3 - 7 b^2 c - 7 b c^2 + 3 c^3) - 4 a^2 (b^4 - b^3 c - 2 b^2 c^2 - b c^3 + c^4)) : .... : ...
 = lies on these lines : {9, 119}, {11, 8257}, {57, 5856}, {100, 12848}, {527, 1155}, {528, 18391}, {1317, 15185}, {1376, 5851}.
= midpoint of X(100) and X(12848).
= reflection of X(11) in X(8257).
    (6 - 9 - 13) - search numbers  of Q9 : (1.30929286950241, -0.531539141891952, 3.40436410190846).
     
     
***    P = X (10)  ---> Q10 =  MIDPOINT OF X(99) AND X(1046)

 

(2 a^3 - b^3 - c^3 + a^2 (b + c) - a (b^2 + c^2)) (a^5 + a^4 (b + c) - b^2 c^2 (b + c) - a^3 (b^2 + c^2) + a (b^4 - b^2 c^2 + c^4)) : .... : ...
= lies on these lines : {99, 1046}, {114, 124}, {115, 8258}, {758, 11711}, {896, 1281}, {4697, 5988}, {5429, 7983}, {10026, 17770}.
= midpoint of X(99) and X(1046).
= reflection of X(115) in X(8258).
    (6 - 9 - 13) - search numbers  of Q10 : (1.62614271745754, -0.297583191784076, 3.09615620662372).
     
     
***   P = X (17) ---> Q17 =  REFLECTION OF X(115) IN X(8259)

 

(6S (2 a^2-b^2-c^2)+Sqrt[3](6 a^4-a^2 (b^2+c^2)+b^4-6 b^2 c^2+c^4))(6S (2 a^2-b^2-c^2)+Sqrt[3](2 a^4-a^2 (b^2+c^2)+(b^2-c^2)^2)) : .... : ...
= lies on these lines :  {114, 6109}, {115, 8259}.

= reflection of X(115 ) in X(8259).
    (6 - 9 - 13) - search numbers  of Q17 : (1.08367436902034, 0.0217358051488212, 3.12545921571811 ).
     
     
***   P = X (18) ---> Q18 =  REFLECTION OF X(115) IN X(8260)

 

(6S (2 a^2-b^2-c^2)-Sqrt[3](6 a^4-a^2 (b^2+c^2)+b^4-6 b^2 c^2+c^4))(6S (2 a^2-b^2-c^2)-Sqrt[3](2 a^4-a^2 (b^2+c^2)+(b^2-c^2)^2)) : .... : ...
= lies on these lines :  {114, 6108}, {115, 8260}.

= reflection of X(115) in X(8260 ).
(6 - 9 - 13) - search numbers  of Q18 : (-18.8800639710005, 10.6579211538638, 4.97597936200199  ).

Angel Montesdeoca