[Kadir Altintas]:
Let ABC be a triangle, P a point and DEF cevian triangle of P. Let Oa,Oaa be the centers of the circles passing through E,F and tangent to circumcircle of ABC. Define Ob,Obb,Oc,Occ cyclically. (*)
(1) For P=X(2) and P=X(4) the perspector of OaObOc and OaaObbOcc is X(5)
(2) For P=X(3) and P=X(5) the triangles OaObOc and OaaObbOcc are perspective but perspector doesn't lie on the Euler line of ABC
Is there any other points such that perspector of OaObOc and OaaObbOcc is on the Euler line of ABC ?
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[Ercole Suppa]
Generalization (Ercole Suppa) Let ABC be a triangle, P a point and DEF cevian triangle of P. Let Oa, Oaa be the centers of the circles passing through E,F and tangent to circumcircle of ABC. Define Ob,Obb,Oc,Occ cyclically.
(1) the triangles OaObOc and OaaObbOcc are perspective for each point P.
(2) If P(u:v:w) are the barycentrics coordinates of P, the perspector Q(P) has barycentric coordinates:
Q(P)=-(b^2-c^2) u^2 (v+w)^2 (b^2 (u+v) w-c^2 v (u+w))-a^4 v (u+v) w (u+w) (-2 v w+u (v+w))+a^2 u (b^2 (u+v) w (v w (-v+w)+u (2 v^2+3 v w-w^2)) +c^2 v (u+w) (v (v-w) w+u (-v^2+3 v w+2 w^2))) :: (barys)
(3) Pairs {P=X(i),Q=X(j)} for these {i,j}: {{1,8143},{2,5},{4,5},{6,8152},{7,1},{8,1158},{69,5894},{80,14000},{99,8151},{189,1158},{190,5592},{253,5894},{264,8146},{598,8145}{8046,14000},{13485,8151}
(4) The perspector Q(P) lies on Euler line if P=X(i) for these i: {2, 4, 671, 2992, 2993, 13574, 19776, 19777}
Therefore we have six new points on Euler line Q(X(671)), Q(X(2992)), Q(X(2993)), Q(X(13574)), Q(X(19776)), Q(X(19777))
(5) Properties of some points Q(X(i)):
*** Q(X(3)) = X(54)X(74) ∩ X(216)X(8612)
= 7*cos(A)+37*cos(3*A)+(-6-107*cos(2*A)+7*cos(6*A))*cos(B-C)+(67*cos(A)+7*cos(3*A)-7*cos(5*A))*cos(2(B-C))+(1-7*cos(2*A)-7*cos(4*A))*cos(3(B-C)) : : (trilinears)
= a^2 (a^18 b^2-6 a^16 b^4+13 a^14 b^6-7 a^12 b^8-21 a^10 b^10+49 a^8 b^12-49 a^6 b^14+27 a^4 b^16-8 a^2 b^18+b^20+a^18 c^2-10 a^16 b^2 c^2+30 a^14 b^4 c^2 -39 a^12 b^6 c^2+30 a^10 b^8 c^2-41 a^8 b^10 c^2+70 a^6 b^12 c^2-65 a^4 b^14 c^2+29 a^2 b^16 c^2-5 b^18 c^2-6 a^16 c^4+30 a^14 b^2 c^4-54 a^12 b^4 c^4 +39 a^10 b^6 c^4+a^8 b^8 c^4-28 a^6 b^10 c^4+36 a^4 b^12 c^4-25 a^2 b^14 c^4+7 b^16 c^4+13 a^14 c^6-39 a^12 b^2 c^6+39 a^10 b^4 c^6-18 a^8 b^6 c^6 +7 a^6 b^8 c^6+9 a^4 b^10 c^6-19 a^2 b^12 c^6+8 b^14 c^6-7 a^12 c^8+30 a^10 b^2 c^8+a^8 b^4 c^8+7 a^6 b^6 c^8-14 a^4 b^8 c^8+23 a^2 b^10 c^8-40 b^12 c^8 -21 a^10 c^10-41 a^8 b^2 c^10-28 a^6 b^4 c^10+9 a^4 b^6 c^10+23 a^2 b^8 c^10+58 b^10 c^10+49 a^8 c^12+70 a^6 b^2 c^12+36 a^4 b^4 c^12-19 a^2 b^6 c^12 -40 b^8 c^12-49 a^6 c^14-65 a^4 b^2 c^14-25 a^2 b^4 c^14+8 b^6 c^14+27 a^4 c^16+29 a^2 b^2 c^16+7 b^4 c^16-8 a^2 c^18-5 b^2 c^18+c^20) : : (barys)
= lies on these lines: {54,74},{216,8612},{6759,10979},{12162,18464},{18383,18416}
=(6-9-13) ETC- search numbers [-22.9847441989530628, 9.02852129800992246, 7.99849321357123101]
*** Q(X(671)) = -66*cos(A)+24*cos(3*A)-21*cos(5*A)+3*cos(7*A)+(9-84*cos(2*A)-cos(6*A))*cos(B-C)+(18*cos(A)-29*cos(3*A)+3*cos(5*A))*cos(2*(B-C))+(-12+33*cos(2*A) -9*cos(4*A))*cos(3*(B-C))+(3*cos(A)+cos(3*A))*cos(4*(B-C)) :: (trilinears)
= 6 a^10-5 a^8 b^2+2 a^6 b^4+6 a^4 b^6-8 a^2 b^8-b^10-5 a^8 c^2-10 a^6 b^2 c^2+3 a^4 b^4 c^2+25 a^2 b^6 c^2-b^8 c^2+2 a^6 c^4+3 a^4 b^2 c^4-42 a^2 b^4 c^4 +2 b^6 c^4+6 a^4 c^6+25 a^2 b^2 c^6+2 b^4 c^6-8 a^2 c^8-b^2 c^8-c^10 :: (barys)
= lies on this line: {2,3}
=(6-9-13) ETC- search numbers [43.9337194814714952, 42.9483012972698179, -46.3698761768815695]
*** Q(X(2992)) = 4 sqrt(3) a^10-7 sqrt(3) a^8 b^2-2 sqrt(3) a^6 b^4+8 sqrt(3) a^4 b^6-2 sqrt(3) a^2 b^8-Sqrt[3] b^10-7 sqrt(3) a^8 c^2+20 sqrt(3) a^6 b^2 c^2-8 sqrt(3) a^4 b^4 c^2 -8 sqrt(3) a^2 b^6 c^2+3 sqrt(3) b^8 c^2-2 sqrt(3) a^6 c^4-8 sqrt(3) a^4 b^2 c^4+20 sqrt(3) a^2 b^4 c^4-2 sqrt(3) b^6 c^4+8 sqrt(3) a^4 c^6-8 sqrt(3) a^2 b^2 c^6 -2 sqrt(3) b^4 c^6-2 sqrt(3) a^2 c^8+3 sqrt(3) b^2 c^8-sqrt(3) c^10-2 a^6 b^2 S+6 a^4 b^4 S-6 a^2 b^6 S+2 b^8 S-2 a^6 c^2 S+4 a^4 b^2 c^2 S+6 a^2 b^4 c^2 S-8 b^6 c^2 S +6 a^4 c^4 S+6 a^2 b^2 c^4 S+12 b^4 c^4 S-6 a^2 c^6 S-8 b^2 c^6 S+2 c^8 S ::
= lies on this line: {2,3}
=(6-9-13) ETC- search numbers [-244.004273991875677, -244.230104478955277, 285.340401732819114]
*** Q(X(2993)) = 4 sqrt(3) a^10-7 sqrt(3) a^8 b^2-2 sqrt(3) a^6 b^4+8 sqrt(3) a^4 b^6-2 sqrt(3) a^2 b^8-sqrt(3) b^10-7 sqrt(3) a^8 c^2+20 sqrt(3) a^6 b^2 c^2-8 sqrt(3) a^4 b^4 c^2 -8 sqrt(3) a^2 b^6 c^2+3 sqrt(3) b^8 c^2-2 sqrt(3) a^6 c^4-8 sqrt(3) a^4 b^2 c^4+20 sqrt(3) a^2 b^4 c^4-2 sqrt(3) b^6 c^4+8 sqrt(3) a^4 c^6-8 sqrt(3) a^2 b^2 c^6 -2 sqrt(3) b^4 c^6-2 sqrt(3) a^2 c^8+3 sqrt(3) b^2 c^8-sqrt(3) c^10+2 a^6 b^2 S-6 a^4 b^4 S+6 a^2 b^6 S-2 b^8 S+2 a^6 c^2 S-4 a^4 b^2 c^2 S-6 a^2 b^4 c^2 S+8 b^6 c^2 S -6 a^4 c^4 S-6 a^2 b^2 c^4 S-12 b^4 c^4 S+6 a^2 c^6 S+8 b^2 c^6 S-2 c^8 S : : (barys)
= lies on this line: {2,3}
=(6-9-13) ETC- search numbers [26.9481681688868384, 26.0075582979273290, -26.8021842638361587]
*** Q(X(13574)) = -66*cos(A)+24*cos(3*A)-21*cos(5*A)+3*cos(7*A)+(9-84*cos(2*A)-*cos(6*A))*cos(B-C)+(18*cos(A)-29*cos(3*A)+3*cos(5*A))*cos(2*(B-C)) +(-12+33*cos(2*A)-9*cos(4*A))*cos(3*(B-C))+(3*cos(A)+*cos(3*A))*cos(4*(B-C)) :: (trilinears)
= 6 a^10-5 a^8 b^2+2 a^6 b^4+6 a^4 b^6-8 a^2 b^8-b^10-5 a^8 c^2-10 a^6 b^2 c^2+3 a^4 b^4 c^2+25 a^2 b^6 c^2-b^8 c^2+2 a^6 c^4 +3 a^4 b^2 c^4-42 a^2 b^4 c^4+2 b^6 c^4+6 a^4 c^6+25 a^2 b^2 c^6+2 b^4 c^6-8 a^2 c^8-b^2 c^8-c^10 : : (barys)
= lies on this line: {2,3}
=(6-9-13) ETC- search numbers [43.9337194814714952, 42.9483012972698179, -46.3698761768815695]
*** Q(X(19776)) = 6 a^16-16 a^14 b^2-2 a^12 b^4+42 a^10 b^6-40 a^8 b^8+4 a^6 b^10+6 a^4 b^12+2 a^2 b^14-2 b^16-16 a^14 c^2+46 a^12 b^2 c^2-21 a^10 b^4 c^2-52 a^8 b^6 c^2+52 a^6 b^8 c^2 +12 a^4 b^10 c^2-31 a^2 b^12 c^2+10 b^14 c^2-2 a^12 c^4-21 a^10 b^2 c^4+72 a^8 b^4 c^4-56 a^6 b^6 c^4-54 a^4 b^8 c^4+81 a^2 b^10 c^4-20 b^12 c^4+42 a^10 c^6 -52 a^8 b^2 c^6-56 a^6 b^4 c^6+72 a^4 b^6 c^6-52 a^2 b^8 c^6+22 b^10 c^6-40 a^8 c^8+52 a^6 b^2 c^8-54 a^4 b^4 c^8-52 a^2 b^6 c^8-20 b^8 c^8+4 a^6 c^10+12 a^4 b^2 c^10 +81 a^2 b^4 c^10+22 b^6 c^10+6 a^4 c^12-31 a^2 b^2 c^12-20 b^4 c^12+2 a^2 c^14+10 b^2 c^14-2 c^16+4 sqrt(3) a^14 S-16 sqrt(3) a^12 b^2 S+20 sqrt(3) a^10 b^4 S -20 sqrt(3) a^6 b^8 S+16 sqrt(3) a^4 b^10 S-4 sqrt(3) a^2 b^12 S-16 sqrt(3) a^12 c^2 S+44 sqrt(3) a^10 b^2 c^2 S-38 sqrt(3) a^8 b^4 c^2 S+10 sqrt(3) a^6 b^6 c^2 S -2 sqrt(3) a^4 b^8 c^2 S+2 sqrt(3) a^2 b^10 c^2 S+20 sqrt(3) a^10 c^4 S-38 sqrt(3) a^8 b^2 c^4 S+4 sqrt(3) a^6 b^4 c^4 S-14 sqrt(3) a^4 b^6 c^4 S+28 sqrt(3) a^2 b^8 c^4 S +10 sqrt(3) a^6 b^2 c^6 S-14 sqrt(3) a^4 b^4 c^6 S-52 sqrt(3) a^2 b^6 c^6 S-20 sqrt(3) a^6 c^8 S-2 sqrt(3) a^4 b^2 c^8 S+28 sqrt(3) a^2 b^4 c^8 S+16 sqrt(3) a^4 c^10 S +2 sqrt(3) a^2 b^2 c^10 S-4 sqrt(3) a^2 c^12 S : : (barys)
= lies on this line: {2,3}
=(6-9-13) ETC- search numbers [-244.004273991875677, -244.230104478955277, 285.340401732819114]
*** Q(X(19777)) = -6 a^16+16 a^14 b^2+2 a^12 b^4-42 a^10 b^6+40 a^8 b^8-4 a^6 b^10-6 a^4 b^12-2 a^2 b^14+2 b^16+16 a^14 c^2-46 a^12 b^2 c^2+21 a^10 b^4 c^2+52 a^8 b^6 c^2-52 a^6 b^8 c^2 -12 a^4 b^10 c^2+31 a^2 b^12 c^2-10 b^14 c^2+2 a^12 c^4+21 a^10 b^2 c^4-72 a^8 b^4 c^4+56 a^6 b^6 c^4+54 a^4 b^8 c^4-81 a^2 b^10 c^4+20 b^12 c^4-42 a^10 c^6+52 a^8 b^2 c^6 +56 a^6 b^4 c^6-72 a^4 b^6 c^6+52 a^2 b^8 c^6-22 b^10 c^6+40 a^8 c^8-52 a^6 b^2 c^8+54 a^4 b^4 c^8+52 a^2 b^6 c^8+20 b^8 c^8-4 a^6 c^10-12 a^4 b^2 c^10-81 a^2 b^4 c^10 -22 b^6 c^10-6 a^4 c^12+31 a^2 b^2 c^12+20 b^4 c^12-2 a^2 c^14-10 b^2 c^14+2 c^16+4 sqrt(3) a^14 S-16 sqrt(3) a^12 b^2 S+20 sqrt(3) a^10 b^4 S-20 sqrt(3) a^6 b^8 S +16 sqrt(3) a^4 b^10 S-4 sqrt(3) a^2 b^12 S-16 sqrt(3) a^12 c^2 S+44 sqrt(3) a^10 b^2 c^2 S-38 sqrt(3) a^8 b^4 c^2 S+10 sqrt(3) a^6 b^6 c^2 S-2 sqrt(3) a^4 b^8 c^2 S +2 sqrt(3) a^2 b^10 c^2 S+20 sqrt(3) a^10 c^4 S-38 sqrt(3) a^8 b^2 c^4 S+4 sqrt(3) a^6 b^4 c^4 S-14 sqrt(3) a^4 b^6 c^4 S+28 sqrt(3) a^2 b^8 c^4 S+10 sqrt(3) a^6 b^2 c^6 S -14 sqrt(3) a^4 b^4 c^6 S-52 sqrt(3) a^2 b^6 c^6 S-20 sqrt(3) a^6 c^8 S-2 sqrt(3) a^4 b^2 c^8 S+28 sqrt(3) a^2 b^4 c^8 S+16 sqrt(3) a^4 c^10 S+2 sqrt(3) a^2 b^2 c^10 S -4 sqrt(3) a^2 c^12 S : : (barys)
= lies on this line: {2,3}
=(6-9-13) ETC- search numbers [26.9481681688868384, 26.0075582979273290, -26.8021842638361587]
Best regards
Ercole Suppa
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