Let P be a
point and A’B’C’ the pedal triangle of P in ABC.
Denote Ga, Oa, Ha, Na,
La 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
|
