HYACINTHOS 26508

[Le Viet An]:

Let ABC be a triangle and L a line.

Denote:

A', B', C' = the orthogonal projections of A, B, C on L, resp.

Ab, Ac = the orthogonal projections of A' on AB, AC, resp.
Similarly Bc, Ba and Ca, Cb.
The lines AbAc, BcBa, CaCb bound a triangle A"B"C".

The circumcircle of A"B"C" touches the line L.

For L = OH (Euler line), OK, OI, IN, .... which is the center of the circle and the touchpoint?

[César Lozada]:

For L(P) = polar trilinear of P=u:v:w (trilinears), the touchpoint T(P) is:

 

T(P) = (2*b^3*(a^2-b^2+c^2)*a^2*c*v^ 4+a^2*b^2*(a^4-2*(b^2-c^2)*a^ 2+(b^2-c^2)*(b^2+3*c^2))*w*v^ 3-2*a^4*b*c*(-a^2+b^2+c^2)*w^ 2*v^2+a^2*c^2*(a^4+2*(b^2-c^2) *a^2-(b^2-c^2)*(3*b^2+c^2))*w^ 3*v+2*c^3*(a^2+b^2-c^2)*a^2*b* w^4)*u^4+(-a*b^3*(a^4-2*(b-2* c)*(b+2*c)*a^2+(b^2-c^2)*(b^2- 3*c^2))*w*v^4-a*b^2*c*(5*a^2- b^2-5*c^2)*(a^2-b^2+c^2)*w^2* v^3-a*b*c^2*(5*a^2-5*b^2-c^2)* (a^2+b^2-c^2)*w^3*v^2-a*c^3*( a^4+2*(2*b-c)*(2*b+c)*a^2+(b^ 2-c^2)*(3*b^2-c^2))*w^4*v)*u^ 3+(b^3*c*(3*a^4-4*(b^2-2*c^2)* a^2+(b^2-c^2)^2)*w^2*v^4+2*b^ 2*c^2*(3*a^4-2*(b^2+c^2)*a^2-( b^2-c^2)^2)*w^3*v^3+b*c^3*(3* a^4+4*(2*b^2-c^2)*a^2+(b^2-c^ 2)^2)*w^4*v^2)*u^2+(-2*a*b^2* c^3*(a^2+b^2-c^2)*w^4*v^3-2*a* b^3*c^2*(a^2-b^2+c^2)*w^3*v^4) *u : : (trilinears)

 

ETC pairs (P,T(P)): (648,3154), (2394,3258), (2395,2679), (2396,2679), (2397,3259), (2398,1566), (2399,10017), (2400,1566), (2401,3259), (2402,5519), (2403,5516), (2406,10017), (2407,3258), (2414,5519), (2415,5516)

or,

ETC-pairs (L(P), T(P)): (({2, 3}, 3154), ({125, 523}, 3258), ({115, 512}, 2679), ({114, 325}, 2679), ({119, 517}, 3259), ({118, 516}, 1566), ({124, 522}, 10017), ({116, 514}, 1566), ({11, 513}, 3259), ({3309, 4904}, 5519), ({3667, 3756}, 5516), ({117, 515}, 10017), ({30, 113}, 3258), ({120, 518}, 5519), ({121, 519}, 5516)

 

T(P) for some notable lines:

 

For L(190 ) = IG = {1, 2} = Nagel line

T =  (3*a-b-c)*(a^3-4*(b+c)*a^2-(3* b^2-19*b*c+3*c^2)*a+(b+c)*(2* b^2-7*b*c+2*c^2))*(b-c)^2 : : (barys)

= 16*X(3633)+X(5205) = 7*X(3756)+10*X(5510)

= On lines: {1, 2}, {3667, 3756}

= midpoint of X(3756) and X(5516)

= [ -1.931084865763773, 1.44888228444231, 3.528862069184519 ]

 

For L(110) = OK = {3,6} = Brocard axis

T = a^2*(b^2-c^2)^2*(a^8-3*(b^2+c^ 2)*a^6+(3*b^4+b^2*c^2+3*c^4)* a^4-(b^4-c^4)*(b^2-c^2)*a^2-b^ 2*c^2*(b^4+c^4)) : : (barys)

= On lines: {2, 12833}, {3, 6}, {115, 512}, {2698, 13137}, {3124, 3288}, {5663, 5915}, {6785, 7737}

= midpoint of X(2698) and X(13137)

= complement of X(12833)

= {X(2024), X(4289)}-Harmonic conjugate of X(8554)

= [ 2.542083945495685, 2.66823532523566, 0.620155128053988 ]

 

For L(476) = {6, 13, 14} = Fermat axis

T= (b^2-c^2)^2*(2*a^2-b^2-c^2)*( 4*a^14-7*(b^2+c^2)*a^12+10*b^ 2*c^2*a^10+2*(b^2+c^2)*(3*b^4- b^2*c^2+3*c^4)*a^8-2*(b^4+c^4) *(3*b^4+4*b^2*c^2+3*c^4)*a^6+( b^6+c^6)*(3*b^4+7*b^2*c^2+3*c^ 4)*a^4+2*(b^12+c^12-(7*b^8+7* c^8-b^2*c^2*(8*b^4-7*b^2*c^2+ 8*c^4))*b^2*c^2)*a^2+(b^4-c^4) *(b^2-c^2)*(-2*b^8-2*c^8+b^2* c^2*(6*b^4-5*b^2*c^2+6*c^4))) : : (barys)

= On lines: {6, 13}, {690, 2682}

= [ 1.353953622880329, 2.04580645676917, 1.599435262814805 ]

 

For L(651) = IGe = {1,7} = Soddy line

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

= On lines: {1, 3}, {11, 513}, {244, 1459}, {1364, 14027}, {3328, 5577}, {6073, 10956}

= midpoint of X(i) and X(j) for these {i,j}: {11, 3025}, {3259, 3937}

= [ 2.990869369979302, 2.76421561005426, 0.346575504187137 ]

 

For L(658) = {1, 3} = IO line

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

= On lines: {1, 7}, {116, 514}

=[ 3.281460570359955, 3.19802937615531, -0.087876118673890 ]

 

For L(107) = HK = Van Aubel line

T = (b^2-c^2)^2*(-a^2+b^2+c^2)*(5* a^14-6*(b^2+c^2)*a^12-(3*b^4- 7*b^2*c^2+3*c^4)*a^10+(b^2+c^ 2)*(2*b^4+3*b^2*c^2+2*c^4)*a^ 8-(b^4+c^4)*(b^2+c^2)^2*a^6+2* (b^4-c^4)*(b^2-c^2)*(3*b^4+b^ 2*c^2+3*c^4)*a^4-(b^6-c^6)*(b^ 2-c^2)*(b^4+6*b^2*c^2+c^4)*a^ 2-(b^4-c^4)*(b^2-c^2)^3*(2*b^ 4+b^2*c^2+2*c^4)) : : (barys)

= On lines: {4, 6}, {127, 525}

= [ 2.532217099771537, 3.34047820630996, 0.159310139182930 ]

 

César Lozada