HYACINTHOS 28066

 

[Antreas P. Hatzipolakis]:

 

 

Let ABC be a triangle, L a line passing through O, P a point on L and A'B'C' the pedal triangle of P.

Denote:

D = the reflection point of L
( = the perspector of ABC, A*B*C*, where A*B*C* is the triangle bounded by the reflections of L in BC, CA, AB, resp.)

 

La, Lb, Lc = the Euler lines of DBC, DCA, DAB, resp.

Conjecture:

The reflections of La,Lb,Lc in PA', PB', PC', resp. are concurrent.

For L =  Euler line and P = H  See:  (Tran Quang Hung and Angel Montesdeoca)
For L = OI line and P = I   See Hyacinthos 28062

Special Case (if conjecture is true):

L = Euler line, D = X(110).

Which is the locus of the point of concurrence as P moves on the Euler line L?

 

 

[César Lozada]:

 

 

Algebraic proof of the conjecture: Very difficult.

 

·         Special case: L=Euler line

 

For P such that OP/OH=t, the point of concurrence Q(P) lies on the line X(3)X(21269) and

OQ /OX(21269) = t

 

ETC pairs (P,Q(P)) : (3,3), (4,21269)

 

Some others:

Q( X(2) ) = midpoint of X(5627) and X(14643)

= 3*S^4+(3*R^2*(81*R^2-6*SA-34*SW)+4*SA^2-SB*SC+11*SW^2)*S^2+(27*R^2*(9*R^2-4*SW)+11*SW^2)*SB*SC : :  (barys)

= 2*X(3)+X(21269), X(1553)+2*X(20379), 2*X(3154)+X(18319), X(3258)-4*X(15088), 4*X(3628)-X(14934), X(7471)+2*X(11801), 2*X(11657)+X(18572), X(16340)-4*X(20304)

= on lines: {3, 21269}, {5, 523}, {30, 14644}, {1553, 20379}, {3154, 18319}, {3258, 15088}, {3628, 14934}, {5627, 14643}, {7471, 11801}, {11657, 18572}, {16340, 20304}

= midpoint of X(5627) and X(14643)

= [ 0.3971914172692749, -0.1612469501146442, 3.5689778701702370 ]

 

Q(X(5)) = midpoint of X(3) and X(21269)

= 3*S^4+(3*R^2*(27*R^2-6*SA-10*SW)+4*SA^2-SB*SC+3*SW^2)*S^2+(45*R^2*(9*R^2-4*SW)+19*SW^2)*SB*SC : : (barys)

= 3*X(5)-X(14934), 3*X(5627)+X(7728), X(14508)-5*X(15027), 3*X(14644)-X(16340), X(14989)+3*X(15061)

= on lines: {3, 21269}, {5, 14385}, {30, 125}, {523, 546}, {1539, 6070}, {5627, 7728}, {7687, 16168}, {8901, 11563}, {14508, 15027}, {14644, 16340}, {14989, 15061}

= midpoint of X(i) and X(j) for these {i,j}: {3, 21269}, {1539, 6070}

= [ -2.7953943211942600, -3.1893458885376690, 7.1388551683693460 ]

 

Q(X(20)) = midpoint of X(3) and X(21269)

= 3*S^4+(-3*R^2*(135*R^2+6*SA-62*SW)+4*SA^2-SB*SC-21*SW^2)*S^2+(99*R^2*(9*R^2-4*SW)+43*SW^2)*SB*SC : : (barys)

= on lines: {3, 21269}, {30, 110}, {523, 15704}

= [ 25.9378773249775600, 24.0635445572695400, -24.9900405154226500 ]

 

Some other remarkable lines were tried (IO, Brcard axis). Concurrence occurs, but results are not nice.

 

César Lozada