HYACINTHOS 27552

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and P a point.

Denote:

A' = the 2nd intersection of the circle (P, PH) and AH
B' = the 2nd intersection of the circle (P, PH) and BH
C' = the 2nd intersection of the circle (P, PH) and CH

[or, to avoid circles:
Pa, Pb, Pc = the orthogonal projections of P on AH, BH, CH, resp.
A', B', C' = the reflections of H in Pa, Pb, Pc, resp.]

Ab, Ac = the orthogonal projections of A' on AB, AC, resp.

La = the Euler line or AAbAc.
Similarly Lb, Lc.

Which is the locus of P such that La, Lb, Lc are concurrent?
The Euler line + something else ?
And which is the locus of the point of concurrence?

[César Lozada]:

 

 

Locus = {Linf} ∪ {Euler line}

 

If OP/OH=t then the point of intersection Q(P) lies on the line X(125)X(403), through ETC’s 125, 403, 6000, 12292, 13399, and

X(125)*Q/X(125)X(403) = (OH/R)^2*(t-1)

 

ETC pairs (P,Q): (4,125), (382,12292), (3153,13399)

 

Some others Q(P):

 

Q(X(2)) = X(74)X(184) ∩ X(125)X(403)

= (SB+SC)*((-6*R^2+5*SA+SW)*S^2+ (216*R^4-9*R^2*(3*SA+7*SW)+5*S A^2-5*SB*SC+4*SW^2)*SA) : : (barys)

= 5*X(125)-2*X(12292), 2*X(185)+X(10990), 5*X(185)-2*X(13148), 4*X(185)-X(14448), 4*X(974)-X(13202), 5*X(10990)+4*X(13148), 2*X(10990)+X(14448), X(11381)-4*X(16270), 8*X(13148)-5*X(14448), 2*X(13491)+X(16003)

= on lines: {51, 974}, {74, 184}, {125, 403}, {185, 1205}, {542, 15072}, {549, 5642}, {1154, 16111}, {2777, 5890}, {2935, 11402}, {3819, 12825}, {7687, 11455}, {7729, 9971}, {11381, 16270}, {13491, 16003}, {14855, 16163}, {15054, 15462}

= reflection of X(51) in X(974)

= {X(185), X(10990)}-Harmonic conjugate of X(14448)

= [ 15.0847616889502900, 15.8297465474735500, -14.2805885227820600 ]

 

Q(X(3)) = X(3)X(74) ∩ X(125)X(403)

= (SB+SC)*((-5*R^2+3*SA+SW)*S^2+ (108*R^4-R^2*(15*SA+32*SW)+3*S A^2-3*SB*SC+2*SW^2)*SA) : : (barys)

= 3*X(4)-4*X(11746), 3*X(74)-X(12281), X(110)-3*X(15072), 3*X(185)-X(13417), 3*X(974)-2*X(11746), 2*X(1112)-3*X(5890), 2*X(1539)-3*X(16222), 3*X(1986)-2*X(13417), 3*X(5890)-X(10721), 3*X(6241)+X(12281), 4*X(6723)-3*X(15030), 3*X(11459)-4*X(13416), X(12111)-3*X(15055), 2*X(12358)-3*X(15055), 4*X(13630)-3*X(16222)

= on lines: {3, 74}, {4, 974}, {24, 9934}, {125, 403}, {184, 13293}, {185, 1986}, {186, 15647}, {265, 11457}, {378, 13198}, {382, 12236}, {389, 13202}, {1112, 5890}, {1181, 2935}, {1204, 13289}, {1539, 13630}, {2071, 3047}, {2693, 10419}, {2781, 6776}, {3028, 9627}, {3520, 11598}, {5504, 11413}, {5656, 7505}, {6699, 12162}, {6723, 15030}, {7592, 15472}, {7687, 11381}, {7728, 12824}, {7731, 13148}, {9786, 17812}, {9826, 10574}, {10018, 14862}, {10117, 10605}, {10264, 15761}, {10293, 10938}, {10575, 17702}, {10619, 10628}, {10706, 16072}, {10733, 12279}, {11455, 12099}, {11806, 12295}, {11807, 13382}, {12133, 12290}, {12140, 16659}, {12283, 12284}, {13754, 16111}, {15059, 15305}

= midpoint of X(i) and X(j) for these {i,j}: {10293, 10938}, {10733, 12279}

= reflection of X(i) in X(j) for these (i,j): (4, 974), (382, 12236), (1539, 13630), (7728, 14708), (7731, 13148), (11455, 12099), (11807, 13382), (12133, 16270)

= 2nd Droz-Farny circle-inverse-of X(74)

= X(974) of anti-Euler triangle

= X(12825) of ABC-X3 reflections triangle

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (1181, 2935, 15463), (1539, 13630, 16222), (5890, 10721, 1112), (6241, 15072, 11456), (7728, 14708, 12824), (10620, 13171, 74), (12111, 15055, 12358), (12133, 16270, 14644), (12290, 14644, 12133)

= [ 20.8820066825843300, 22.2047506268805900, -21.3697044209718700 ]

 

Q(X(5)) = X(54)X(74) ∩ X(125)X(403)
= (SB+SC)*((-R^2+2*SA)*S^2+(108* R^4-R^2*(12*SA+31*SW)+2*SA^2-2 *SB*SC+2*SW^2)*SA) : : (barys)

= 3*X(51)-X(10721), 3*X(74)+X(7722), 3*X(125)-X(12292), X(146)-5*X(10574), X(146)-3*X(16223), 3*X(185)-X(7722), 3*X(376)+X(12284), 3*X(389)-2*X(1112), 5*X(389)-2*X(16105), 3*X(974)-X(1112), 4*X(974)-X(11807), 5*X(974)-X(16105), 4*X(1112)-3*X(11807), 5*X(1112)-3*X(16105), 2*X(5972)-3*X(16836), 5*X(10574)-3*X(16223), 3*X(11800)-4*X(13358), 3*X(11806)-2*X(13358), 5*X(11807)-4*X(16105)

= on lines: {3, 17838}, {30, 11800}, {51, 10721}, {54, 74}, {67, 13623}, {113, 9729}, {125, 403}, {140, 5663}, {146, 10574}, {182, 5621}, {265, 10575}, {376, 12284}, {389, 974}, {511, 16111}, {578, 2935}, {1539, 5462}, {1598, 17812}, {1614, 17701}, {1986, 10990}, {2807, 11709}, {3448, 15072}, {3522, 12273}, {5449, 10264}, {5562, 15055}, {5622, 12294}, {5890, 12244}, {6102, 14677}, {7687, 13474}, {7728, 9730}, {7729, 10117}, {9786, 9919}, {10110, 13202}, {10113, 14915}, {10605, 13171}, {11381, 14644}, {11557, 13630}, {11558, 11801}, {11793, 12825}, {12041, 13754}, {12121, 14855}, {12162, 15061}, {12219, 15021}, {12236, 13598}, {15012, 16222}, {15030, 15059}

= midpoint of X(i) and X(j) for these {i,j}: {265, 10575}, {1986, 10990}, {6102, 14677}

= reflection of X(i) in X(j) for these (i,j): (113, 9729), (389, 974), (1539, 5462), (1986, 13382), (7687, 16270), (11557, 13630)

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (74, 13198, 13293), (146, 10574, 16223), (5890, 12244, 13417), (13198, 13293, 11430)

= [ 12.1861391921332800, 12.6422445077700400, -10.7360305736871600 ]

 

Q(X(20)) = X(74)X(6759) ∩ X(125)X(403)

= (SB+SC)*((-14*R^2+7*SA+3*SW)*S ^2+(216*R^4-R^2*(33*SA+65*SW)+ 7*SA^2-7*SB*SC+4*SW^2)*SA) : : (barys)

= 3*X(125)-2*X(12292), 3*X(185)-2*X(1112), 5*X(974)-4*X(15465), 4*X(1112)-3*X(13202), 2*X(5972)-3*X(15072), 3*X(6241)-X(7722), 5*X(6241)-X(7731), 4*X(6241)-X(14448), 4*X(6723)-3*X(15305), 5*X(7722)-3*X(7731), 4*X(7722)-3*X(14448), 4*X(7731)-5*X(14448), 5*X(11381)-8*X(15465), 3*X(12295)-4*X(13358)

= on lines: {74, 6759}, {113, 13491}, {125, 403}, {185, 1112}, {550, 5562}, {974, 11381}, {1204, 9934}, {2777, 6241}, {2935, 12174}, {3047, 13445}, {5972, 15072}, {6723, 15305}, {7687, 12290}, {10620, 12893}, {11456, 13293}, {11598, 13367}, {12295, 13358}

= reflection of X(113) in X(13491)

= [ 38.2737416634864200, 41.3297628651016900, -42.6370521155412900 ]

 

César Lozada