Pedal triangles and some minimizers

Let P be a point and A’B’C’ the pedal triangle of P in ABC.

Denote  G_a, O_a, H_a, N_a, L_a the centroid, circumcenter, orthocenter, NPC and De Longchamps point of AB’C’, respectively, and similarly for triangles BC’A’, CA’B’. 

Denote G*_a, O*_a, H*_a, N*_a, L*_a the centroid, circumcenter, orthocenter, NPC and De Longchamps point of PB’C’, respectively, and similarly for triangles PC’A’, PA’B’. 

The sums of squares of the distances from these points to A, A’, P has been required to be minimal and the  points that minimize them are shown in the following table:

DD to be minimized	P minimizing DD
|AGa|2 + |BGb|2 + |CGc|2	P1 = a*(4*SA^2-32*R^2*SA+3*S^2) On Stammler hyperbola On lines: (1,1407), (6,20), (22,1620), (64,159), (155,550), (170,1740), (195,3534), (220,610),(376,1498), (3387,4096) P=(2*R*r+r^2+s^2)*X(1)-6*R*(4*R+r)*X(2)+(32*R^2-2*r^2)*X(3) ( 14.765340974655620, 14.19256039538952, -12.999727010895580 )
|AOa|2 + |BOb|2 + |COc|2	P = X(2) = G
|AHa|2 + |BHb|2 + |CHc|2	P = X(394) = X(69)-CEVA CONJUGATE OF X(3)
|ANa|2 + |BNb|2 + |CNc|2	P2 = a*((-18*SW+27*R^2)*SA^2+72*SW*R^2*SA+S^2*(-10*SW+7*R^2)) On lines:  (6, 5189), (1495, 3819) ( 13.611601433916640, 12.74655455628163, -11.466150872710580 )
|ALa|2 + |BLb|2 + |CLc|2	P = X(6) = K
|AG*a|2 + |BG*b|2 + |CG*c|2	P3 = a*SA*(8*SA^2+(SW-54*R^2)*SA+8*S^2-144*SW*R^2) ( 5.855422523640938, 4.83444910933685, -2.408764527775413 )
|AO*a|2 + |BO*b|2 + |CO*c|2	P = X(2) = G
|AH*a|2 + |BH*b|2 + |CH*c|2	P1
|AN*a|2 + |BN*b|2 + |CN*c|2	P4 = a*((-50*SW+75*R^2)*SA^2+600*SW*R^2*SA+3*S^2*(-14*SW+13*R^2)) ( 8.125745686328431, 7.18935668022910, -5.086926613479660 )
|AL*a|2 + |BL*b|2 + |CL*c|2	P = X(394) = X(69)-CEVA CONJUGATE OF X(3)
|A’Ga|2 + |B’Gb|2 + |C’Gc|2	P5 = a*(8*SA^2+(-16*R^2-15*SW)*SA+48*S^2) ( -1.479971604569610, -0.87823328755578, 4.931735960016664 )
|A’Oa|2 + |B’Ob|2 + |C’Oc|2	P = X(3060) = INTERSECTION OF LINES X(2)X(51) AND X(4)X(52)
|A’Ha|2 + |B’Hb|2 + |C’Hc|2	P = X(6) = K
|A’Na|2 + |B’Nb|2 + |C’Nc|2	P6 = a*(SA^2-4*SW*SA+17*S^2) On lines: (2,5093), (51,110),  (511,P7), (576,2979) ( -0.361699523684579, 0.30020285948639, 3.599769974732840 )
|A’La|2 + |B’Lb|2 + |C’Lc|2	P = X(3167) = X(6)-CEVA CONJUGATE OF X(3)
|A’G*a|2 + |B’G*b|2 + |C’G*c|2	P = X(6) = K
|A’O*a|2 + |B’O*b|2 + |C’O*c|2	P = X(3060) = INTERSECTION OF LINES X(2)X(51) AND X(4)X(52)
|A’H*a|2 + |B’H*b|2 + |C’H*c|2	P = X(3) = O
|A’N*a|2 + |B’N*b|2 + |C’N*c|2	P7 = a*(SA^2+4*SW*SA+9*S^2) On lines: (2,3167), (23,3796), (182,3060), (511, P6),(575,1994), (1495, 5012), (3410, 3589) ( 2.875410526058825, 2.76353926438645, 0.400332440689689 )
|A’L*a|2 + |B’L*b|2 + |C’L*c|2	P = X(154) = X(3)-CEVA CONJUGATE OF X(6)
|PGa|2 + |PGb|2 + |PGc|2	P8 = a*((54*R^2-SW)*SA^2-16*R^2*SW*SA+144*S^2*R^2) ( 1.512740668269645, 1.28841604860747, 2.050496139823989 )
|POa|2 + |POb|2 + |POc|2	P = X(2) = G
|PHa|2 + |PHb|2 + |PHc|2	P = X(6) = K
|PNa|2 + |PNb|2 + |PNc|2	P9 = a*((75*R^2-2*SW)*SA^2-24*R^2*SW*SA+3*S^2*(2*SW+125*R^2)) ( 1.265145509138911, 1.35115576252445, 2.121335642095649 )
|PLa|2 + |PLb|2 + |PLc|2	P = X(394) = X(69)-CEVA CONJUGATE OF X(3)
|PG*a|2 + |PG*b|2 + |PG*c|2	P = X(6) = K
|PO*a|2 + |PO*b|2 + |PO*c|2	P = X(2) = G
|PH*a|2 + |PH*b|2 + |PH*c|2	P = X(394) = X(69)-CEVA CONJUGATE OF X(3)
|PN*a|2 + |PN*b|2 + |PN*c|2	P10 = a*((-2*SW+27*R^2)*SA^2-8*R^2*SW*SA-3*S^2*(-2*SW+3*R^2)) On lines: (5,323), (1994,3292) ( 2.321361156037366, 0.67831187364855, 2.099666497364295 )
|PL*a|2 + |PL*b|2 + |PL*c|2	P1