HYACINTHOS 27051

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and A'B'C' the pedal triangle of I.

Denote:

A" = AH /\ B'C'
B" = BH /\ C'A'
C" = CH /\ A'B'

The NPCs of A"HI, B"HI, C"HI are coaxial.

2nd, other than the midpoint of HI, intersection?

 

GENERALIZATION

 

Let ABC be a triangle, P a point  and A'B'C' the pedal triangle of I.

Denote:

A" = AP /\ B'C'
B" = BP /\ C'A'
C" = CP /\ A'B'

Which is the locus of P such that the NPCs of A"PI, B"PI, C"PI are coaxial ?
The entire plane?

 

[César Lozada]:

 

>>Which is the locus of P such that the NPCs of A"PI, B"PI, C"PI are coaxial ? The entire plane?

The entire plane – {X(1)}

For P= u:v:w (trilinears), the 2nd point of intersection is:

Q(P) = b*(-c^3+(-2*a+b)*c^2+(2*a^2+b* a+b^2)*c+(a-b)*(a^2+b^2))*u^2* v^2+c*(-b^3+(-2*a+c)*b^2+(2*a^ 2+c*a+c^2)*b+(a-c)*(a^2+c^2))* u^2*w^2-b*c*(2*a^2-(b+c)*a-(b- c)^2)*v^2*w^2-b^3*(a-b+c)*v^4- a^2*((b+c)*a-(b-c)^2)*u^4+a*b* (-a^2+b^2+c^2)*u*v^2*w+c*a*(- a^2+b^2+c^2)*u*v*w^2+(-(b^2+c^ 2)*a^2+(b^2-c^2)^2)*w*u^2*v-2* a*b*u*w^3*c^2-2*a*v^3*b^2*u*c+ c*a*(b^2-c^2+a^2)*v*u^3+c^2*( b^2-c^2+a^2)*w^3*v+b^2*(c^2+a^ 2-b^2)*w*v^3+a*b*(c^2+a^2-b^2) *u^3*w-c^3*(a+b-c)*w^4 : :

 

ETC-pairs: (514,927), (650,14733), (3008,927)

 

Examples:

Q( X(2) ) = incircle-inverse-of X(150)

= 2*a^4-(b+c)*a^3-2*(b^2-b*c+c^ 2)*a^2+(b+c)*(2*b^2-3*b*c+2*c^ 2)*a+(b^3-c^3)*(b-c) : : (barys)

= X(1281)-5*X(3616), 7*X(3622)+X(5992)

= on lines: {1, 147}, {10, 4561}, {86, 99}, {325, 519}, {542, 4909}, {620, 6703}, {1125, 5976}, {1281, 3616}, {1387, 2783}, {1565, 2792}, {2786, 3960}, {2795, 11281}, {3026, 3027}, {3622, 5992}, {3986, 10754}, {4887, 5126}, {4987, 7267}

= midpoint of X(1) and X(5988)

= incircle-inverse-of X(150)

= [ 1.9827995563140580, 2.1699139353847460, 1.2232781160346050 ]

 

Q( X(6) ) = incircle-inverse-of X(14667)

= a*((b+c)*a^7-(b^2+c^2)*a^6-(b^ 3+c^3)*a^5+(b^4+c^4)*a^4-(b^2- c^2)^2*(b+c)*a^3+(b^2-c^2)^2*( b-c)^2*a^2+(b^3-c^3)*(b^4-c^4) *a-(b^4-c^4)*(b^2-c^2)*(b-c)^ 2) : : (barys)

= on lines: {1, 2931}, {6, 10101}, {11, 113}, {65, 74}, {81, 105}, {517, 10149}, {518, 3580}, {526, 676}, {1360, 3024}, {1421, 6126}, {1495, 3827}, {2772, 11028}, {3724, 8758}, {5728, 12904}, {5902, 7986}

= incircle-inverse-of X(14667)

= [ 1.3295572451314970, 1.5206849135101880, 1.9742408132627890 ]

 

César Lozada