HYACINTHOS 25118

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and P a point.

Denote:

O1, O2, O3 = the circumcenters of PBC, PCA, PAB, resp.

Ab, Ac = the orthgonal projections of O1 on the perpendicular bisectors of AC, AB, resp.

Bc, Ba = the orthgonal projections of O2 on the perpendicular bisectors of BA, BC, resp.

Ca, Cb = the orthgonal projections of O3 on the perpendicular bisectors of CB, CA, resp.

La, Lb, Lc = the Euler lines of O1AbAc, O2BcBa, O3CaCb, resp.

A*B*C*  the triangle bounded by La, Lb, Lc, resp.

ABC, A*B*C* are parallelogic.

The parallelogic center (ABC, A*B*C*) is a point on the circumcircle (the antipode of the reflection point of the Euler line = X74)

Which is the locus of P such that La, Lb, Lc are concurrent?

H lies on the locus.

 

[César Lozada]:

 

Locus = {Jerabek hyperbola} \/ { circumcircle}

 

For P on the circumcircle the point of concurrence Z(P)=O.

If P lies on the Jerabek hyperbola then Z(P)=midpoint of {P, X(110)}

 

The locus of Z(P) is the bicevian-conic of {X(2), X(110)}, through ETC’s: 3, 5, 6, 113, 141, 206, 942, 960, 1147, 1209, 1493, 1511, 2574, 2575, 2883, 4550, 5181, 6593, 8542, 10639, 10640, 10960, 10962. Its center is X(5972) and perspector

Q = isogonal conjugate of {6,1632}/\{98,648}

 

ETC pairs (P,Z(P)): (3,1511), (4,113), (6,6593), (74,3284), (265,11062), (2574,6), (2575,6)

 

Z( X(54) ) = midpoint of X(54) and X(110)

= a^3*(3*SA^2-S^2)*(SA^2-3*R^2* SA+S^2-2*SB*SC) : :  (trilinears)

= 4*R^4*X(5)-(7*R^2-2*SW)^2*X( 49)

= On lines: {3,8157}, {5,49}, {6,3200}, {113,10540}, {125,6689}, {184,399}, {186,323}, {195,568}, {215,942}, {539,5642}, {1209,5972}, {1539,10296}, {2070,11557}, {2883,7728}, {3047,5609}, {3165,10639}, {3166,10640}, {3519,10018}, {5012,10264}, {5181,5965}, {5663,10610}, {7502,7731}, {10066,10091}, {10082,10088}

= midpoint of X(54) and X(110)

= reflection of X(i) in X(j) for these (i,j): (125,6689), (1209,5972)

= circumcircle-inverse-of-X( 8157)

= [ -2.640098039183514, 4.92681585348379, 1.448298755272610 ]

 

Z( X(64) ) = midpoint of X(64) and X(110)

= (SA-24*R^2+5*SW)*(SA^2-4*(6*R^ 2-SW)*SA+S^2) : : (trilinears)

= On lines: {3,9934}, {5,1539}, {6,74}, {64,110}, {113,10257}, {125,1885}, {942,2778}, {1112,1204}, {1147,3357}, {1493,10628}, {1503,5181}, {1511,6000}, {1853,10733}, {2883,5972}, {4550,11204}, {5925,10721}, {6640,7728}, {8567,10117}, {10060,10091}, {10076,10088}

= midpoint of X(i) and X(j) for these {i,j}: {64,110}, {74,2935}, {5925,10721}

= reflection of X(i) in X(j) for these (i,j): (125,6696), (2883,5972)

= complementary conjugate of X(403)

= [ 16.267443520070720, 16.15762297464304, -15.053433048262670 ]

 

César Lozada