ETC1

X(73131) = X(4)X(94)∩X(526)X(20304)

Barycentrics    a^2*(a^16*b^4-7*a^14*b^6+21*a^12*b^8-35*a^10*b^10+35*a^8*b^12-21*a^6*b^14+7*a^4*b^16-a^2*b^18-2*a^16*b^2*c^2+6*a^14*b^4*c^2-6*a^12*b^6*c^2+7*a^10*b^8*c^2-15*a^8*b^10*c^2+12*a^6*b^12*c^2+4*a^4*b^14*c^2-9*a^2*b^16*c^2+3*b^18*c^2+a^16*c^4+6*a^14*b^2*c^4-22*a^12*b^4*c^4+22*a^10*b^6*c^4-9*a^8*b^8*c^4+17*a^6*b^10*c^4-41*a^4*b^12*c^4+39*a^2*b^14*c^4-13*b^16*c^4-7*a^14*c^6-6*a^12*b^2*c^6+22*a^10*b^4*c^6-14*a^8*b^6*c^6-9*a^6*b^8*c^6+49*a^4*b^10*c^6-53*a^2*b^12*c^6+18*b^14*c^6+21*a^12*c^8+7*a^10*b^2*c^8-9*a^8*b^4*c^8-9*a^6*b^6*c^8-38*a^4*b^8*c^8+24*a^2*b^10*c^8-3*b^12*c^8-35*a^10*c^10-15*a^8*b^2*c^10+17*a^6*b^4*c^10+49*a^4*b^6*c^10+24*a^2*b^8*c^10-10*b^10*c^10+35*a^8*c^12+12*a^6*b^2*c^12-41*a^4*b^4*c^12-53*a^2*b^6*c^12-3*b^8*c^12-21*a^6*c^14+4*a^4*b^2*c^14+39*a^2*b^4*c^14+18*b^6*c^14+7*a^4*c^16-9*a^2*b^2*c^16-13*b^4*c^16-a^2*c^18+3*b^2*c^18) : :
X(73131) = X[265]+X[70069]

Antreas Hatzipolakis and Ercole Suppa, euclid 10408.

X(73131) lies on these lines: {4, 94}, {520, 72511}, {526, 20304}, {1511, 36829}, {39987, 51231}

X(73131) = midpoint of X(265) and X(70069)
X(73131) = foot of the perpendicular from X(20304) to the line X(4)X(94)
X(73131) = {X(265),X(70069)}-harmonic conjugate of X(5663)

---

X(73132) = X(4)X(94)∩X(526)X(39509)

Barycentrics    a^2*(a^16*b^4-7*a^14*b^6+21*a^12*b^8-35*a^10*b^10+35*a^8*b^12-21*a^6*b^14+7*a^4*b^16-a^2*b^18-2*a^16*b^2*c^2+4*a^14*b^4*c^2+2*a^12*b^6*c^2-3*a^10*b^8*c^2-15*a^8*b^10*c^2+22*a^6*b^12*c^2-4*a^4*b^14*c^2-7*a^2*b^16*c^2+3*b^18*c^2+a^16*c^4+4*a^14*b^2*c^4-22*a^12*b^4*c^4+20*a^10*b^6*c^4+17*a^8*b^8*c^4-19*a^6*b^10*c^4-27*a^4*b^12*c^4+39*a^2*b^14*c^4-13*b^16*c^4-7*a^14*c^6+2*a^12*b^2*c^6+20*a^10*b^4*c^6-50*a^8*b^6*c^6+15*a^6*b^8*c^6+67*a^4*b^10*c^6-65*a^2*b^12*c^6+18*b^14*c^6+21*a^12*c^8-3*a^10*b^2*c^8+17*a^8*b^4*c^8+15*a^6*b^6*c^8-86*a^4*b^8*c^8+34*a^2*b^10*c^8-3*b^12*c^8-35*a^10*c^10-15*a^8*b^2*c^10-19*a^6*b^4*c^10+67*a^4*b^6*c^10+34*a^2*b^8*c^10-10*b^10*c^10+35*a^8*c^12+22*a^6*b^2*c^12-27*a^4*b^4*c^12-65*a^2*b^6*c^12-3*b^8*c^12-21*a^6*c^14-4*a^4*b^2*c^14+39*a^2*b^4*c^14+18*b^6*c^14+7*a^4*c^16-7*a^2*b^2*c^16-13*b^4*c^16-a^2*c^18+3*b^2*c^18) : :
X(73132) = X[4]+X[70069]

Antreas Hatzipolakis and Ercole Suppa, euclid 10408.

X(73132) lies on these lines: {4, 94}, {526, 39509}, {23217, 73022}

X(73132) = midpoint of X(4) and X(70069)
X(73132) = foot of the perpendicular from X(i) to the line X(j)X(k) for these {i,j,k}: {39509, 4, 94}, {4, 39509, 61574}
X(73132) = center of the orthopolar conic of X(70069)
X(73132) = {X(4),X(70069)}-harmonic conjugate of X(5663)

---

X(73133) = X(30)X(1141)∩X(137)X(2070)

Barycentrics    a^22-6*a^20*b^2+15*a^18*b^4-19*a^16*b^6+8*a^14*b^8+14*a^12*b^10-28*a^10*b^12+20*a^8*b^14-a^6*b^16-8*a^4*b^18+5*a^2*b^20-b^22-6*a^20*c^2+25*a^18*b^2*c^2-40*a^16*b^4*c^2+29*a^14*b^6*c^2-9*a^12*b^8*c^2+8*a^10*b^10*c^2-10*a^8*b^12*c^2-15*a^6*b^14*c^2+41*a^4*b^16*c^2-31*a^2*b^18*c^2+8*b^20*c^2+15*a^18*c^4-40*a^16*b^2*c^4+37*a^14*b^4*c^4-13*a^12*b^6*c^4-3*a^8*b^10*c^4+32*a^6*b^12*c^4-80*a^4*b^14*c^4+80*a^2*b^16*c^4-28*b^18*c^4-19*a^16*c^6+29*a^14*b^2*c^6-13*a^12*b^4*c^6+a^10*b^6*c^6+2*a^8*b^8*c^6-19*a^6*b^10*c^6+74*a^4*b^12*c^6-110*a^2*b^14*c^6+55*b^16*c^6+8*a^14*c^8-9*a^12*b^2*c^8+2*a^8*b^6*c^8+6*a^6*b^8*c^8-27*a^4*b^10*c^8+91*a^2*b^12*c^8-62*b^14*c^8+14*a^12*c^10+8*a^10*b^2*c^10-3*a^8*b^4*c^10-19*a^6*b^6*c^10-27*a^4*b^8*c^10-70*a^2*b^10*c^10+28*b^12*c^10-28*a^10*c^12-10*a^8*b^2*c^12+32*a^6*b^4*c^12+74*a^4*b^6*c^12+91*a^2*b^8*c^12+28*b^10*c^12+20*a^8*c^14-15*a^6*b^2*c^14-80*a^4*b^4*c^14-110*a^2*b^6*c^14-62*b^8*c^14-a^6*c^16+41*a^4*b^2*c^16+80*a^2*b^4*c^16+55*b^6*c^16-8*a^4*c^18-31*a^2*b^2*c^18-28*b^4*c^18+5*a^2*c^20+8*b^2*c^20-c^22 : :
X(73133) = 3*X[15392]-2*X[40631], X[35729]-2*X[67090], 2*X[137]-X[2070], 2*X[186]-3*X[57324], X[930]-2*X[37938], 4*X[2072]-3*X[57316], 2*X[10096]-3*X[25147], X[11671]+X[46450], X[13619]-2*X[38618], X[14157]-2*X[43966], 2*X[18403]-X[31656], 4*X[34837]-3*X[37955], 3*X[37943]-4*X[61594], 2*X[38615]-3*X[65085], X[43893]-2*X[73017]

Antreas Hatzipolakis and Ercole Suppa, euclid 10408.

X(73133) lies on these lines: {30, 1141}, {137, 2070}, {186, 57324}, {523, 19552}, {930, 37938}, {2072, 57316}, {3153, 25150}, {3484, 28342}, {5899, 30715}, {7502, 72940}, {10096, 25147}, {11671, 46450}, {13619, 38618}, {14157, 43966}, {18403, 31656}, {34837, 37955}, {37943, 61594}, {38615, 65085}, {43893, 73017}

X(73133) = midpoint of X(11671) and X(46450)
X(73133) = reflection of X(i) in X(j) for these {i,j}: {930, 37938}, {2070, 137}, {13619, 38618}, {14157, 43966}, {31656, 18403}, {35729, 67090}, {43893, 73017}
X(73133) = reflection of X(i) in the line X(j)X(k) for these {i,j,k}: {1157, 5, 523}, {19552, 2, 3}
X(73133) = foot of the perpendicular from X(19552) to the line X(1141)X(1157)
X(73133) = center of circle {X(11671), X(14731), X(46450)}

---

X(73134) = X(30)X(137)∩X(186)X(61594)

Barycentrics    2*a^22-11*a^20*b^2+24*a^18*b^4-24*a^16*b^6+2*a^14*b^8+28*a^12*b^10-42*a^10*b^12+26*a^8*b^14+4*a^6*b^16-17*a^4*b^18+10*a^2*b^20-2*b^22-11*a^20*c^2+46*a^18*b^2*c^2-73*a^16*b^4*c^2+51*a^14*b^6*c^2-15*a^12*b^8*c^2+16*a^10*b^10*c^2-16*a^8*b^12*c^2-37*a^6*b^14*c^2+84*a^4*b^16*c^2-60*a^2*b^18*c^2+15*b^20*c^2+24*a^18*c^4-73*a^16*b^2*c^4+84*a^14*b^4*c^4-42*a^12*b^6*c^4-2*a^10*b^8*c^4+5*a^8*b^10*c^4+67*a^6*b^12*c^4-165*a^4*b^14*c^4+151*a^2*b^16*c^4-49*b^18*c^4-24*a^16*c^6+51*a^14*b^2*c^6-42*a^12*b^4*c^6+26*a^10*b^6*c^6-6*a^8*b^8*c^6-50*a^6*b^10*c^6+161*a^4*b^12*c^6-206*a^2*b^14*c^6+90*b^16*c^6+2*a^14*c^8-15*a^12*b^2*c^8-2*a^10*b^4*c^8-6*a^8*b^6*c^8+32*a^6*b^8*c^8-63*a^4*b^10*c^8+175*a^2*b^12*c^8-96*b^14*c^8+28*a^12*c^10+16*a^10*b^2*c^10+5*a^8*b^4*c^10-50*a^6*b^6*c^10-63*a^4*b^8*c^10-140*a^2*b^10*c^10+42*b^12*c^10-42*a^10*c^12-16*a^8*b^2*c^12+67*a^6*b^4*c^12+161*a^4*b^6*c^12+175*a^2*b^8*c^12+42*b^10*c^12+26*a^8*c^14-37*a^6*b^2*c^14-165*a^4*b^4*c^14-206*a^2*b^6*c^14-96*b^8*c^14+4*a^6*c^16+84*a^4*b^2*c^16+151*a^2*b^4*c^16+90*b^6*c^16-17*a^4*c^18-60*a^2*b^2*c^18-49*b^4*c^18+10*a^2*c^20+15*b^2*c^20-2*c^22 : :
X(73134) = X[186]-2*X[61594], 2*X[2072]-X[38615], X[13619]-3*X[57324], X[18859]+X[44976], 3*X[23516]-2*X[44234]

Antreas Hatzipolakis and Ercole Suppa, euclid 10408.

X(73134) lies on these lines: {30, 137}, {186, 61594}, {523, 72975}, {2072, 38615}, {2937, 72940}, {13619, 57324}, {14071, 70065}, {14106, 38609}, {18400, 43966}, {18403, 25150}, {18859, 44976}, {23516, 44234}, {57584, 71363}

X(73134) = midpoint of X(18859) and X(44976)
X(73134) = reflection of X(i) in X(j) for these {i,j}: {186, 61594}, {38615, 2072}
X(73134) = reflection of X(i) in the line X(j)X(k) for these {i,j,k}: {6150, 5, 523}, {72975, 2, 3}
X(73134) = foot of the perpendicular from X(i) to the line X(j)X(k) for these {i,j,k}: {72975, 137, 6150}

---

A CYCLOLOGIC THEOREM RELATED TO EXCENTRAL TRIANGLE.

[APH]

Excentral version

Let ABC be a triangle, IaIbIc the excentral triangle and P a point..

Denote

Pa, Pb, Pc = same to P points of IaBC, IbCA, IcAB, resp.

ABC, PaPbPc are circumcyclologic

Cyclologic center (ABC, PaPbPc) = Q = ? (on the circumcircle of ABC)
Cyclologic center (PaPbPc, ABC) = Q* = ? (on the circumcircle of PaPbPc)

[Ercole Suppa]

1. P on the Euler Line:

Q = X(100)
Locus of Q* as P moves on the Euler line: K086 2. P on the Brocard axis:

Q = X(101)

[Bernard Gibert]

If P is on a line through X(3) and a strong point M then the locus seems to be a circular cK(#X1,R) with singular focus F.

When M = X2, you get K086.

When M = X6, you get K040.

[APH]

For the case of the cyclologic center (ABC, PaPbPc) M can be any point strong or not.
That is:
Let ABC be a triangle, IaIbIc the excentral triangle, M a fixed point and P a point on the line OM.

Denote:

Pa, Pb,Pc = same to P points of IaBC,IbCA,IcAB, resp.

The triangles ABC, PaPbPc are circumcyclologic.

As P moves on the line OM:
The cyclologic center (ABC, PaPbPc) is a fixed point Q on the circumcircle.
The locus of the cyclologic center (PaPbPc, ABC) is a cubic.

Let's see the Q's. The class of the cubics is a subject of Bernard Gibert.
1. P on the Euler line
Q = X(100) = Reflection point of IO = X(1)X(3) line = Reflection point of Euler line of INTOUCH triangle (pedal triangle of I).

2. P on the Brocard axis
Q = X(101) = Reflection point of X(1)X(7) line = Reflection point of Brocard axis of INTOUCH triangle (pedal triangle of I).

Generalization

Let M be a fixed Point and P be a point on the line OM = L
Denote: P' = the same to P point of the INTOUCH triangle.
L' = the same to L line of the INTOUCH triangle.

Then Q is the reflection point of the L' line = IP' line of ABC

Note:
Reflection point of a line L:
The reflections La, Lb, Lc of L in the siedelines BC, CA, AB, resp. bound a triangle A*B*C*.
ABC, A*B*C* are perspective.The perspector, lying on the circumcircle, is called "Reflection pont of the line L"
It is the incenter (or an excenter) of the triangle A*B*C*.

Case of OM with M = I= X(1)

1. P = X(1) = O of INTOUCH triangle.
Pa, Pb, Pc = X(1) of IaBC, IbCA, IcAB, resp.
Cyclologic center (ABC, PaPbPc) = Q

2. P = X(354) = X(2) of the INTOUCH triangle
Pa, Pb, Pc = X(2) of the intouch triangles of IaBC, IbCA, IcAB, resp.
Cyclologic center (ABC, PaPbPc) = Q1

3. P = X(942) = X(5) of the INTOUCH triangle
Pa, Pb, Pc = X(5) of the intouch triangles of IaBC, IbCA, IcAB, resp.
Cyclologic center (ABC, PaPbPc) = Q2

Q, Q1, Q2 coincide.

Q is the reflection point of the IO line of the INTOUCH triangle.
It is the line passing thrpugh the incenters of ABC and the intouch triangle.

A CYCLOLOGIC THEOREM RELATED TO ORTHIC TRIANGLE

[APH]

Orthic Version

Let ABC be a triangle, HaHbHc the orthic triangle and P a point.

Denote:

Pa, Pb, Pc = same to P points of AHbHc, BHcHa, CHaHb, resp.

HaHbHc, PaPbPc are circumcyclologic

Cyclologic center (HaHbHc, PaPbPc = ? (on the circumcircle of HaHbHc = NPC)
Cyclologic center (PaPbPc, HaHbHc) = ? (on the circumcircle of PaPbPc)

[Ercole Suppa]

cyclologic center (HaHbHc, PaPbPc) = Poncelet point(isogonal conjugate(P))

cyclologic center (PaPbPc, HaHbHc) = orthoassociate(isogonal conjugate(circumcircleInverse(P)))

Euclid 10026

LOCUS PROBLEM

Problem by Antreas Hatzipolakis
Solution by Francisco Javier García Capitán

ETC LISTING OF Q

X(72803)

FOUR POINTS ON THE EULER LINE

Problem by Antreas Hatzipolakis
Solution by Francisco Javier García Capitán
FOUR POINTS ON THE EULER LINE

ETC LISTINGS

X(72698)

X(72699)

X(72700)

X(72701)

EULER

X(72398) = 105TH HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics 6*a^10 - 11*a^8*b^2 - 2*a^6*b^4 + 12*a^4*b^6 - 4*a^2*b^8 - b^10 - 11*a^8*c^2 + 38*a^6*b^2*c^2 - 21*a^4*b^4*c^2 - 9*a^2*b^6*c^2 + 3*b^8*c^2 - 2*a^6*c^4 - 21*a^4*b^2*c^4 + 26*a^2*b^4*c^4 - 2*b^6*c^4 + 12*a^4*c^6 - 9*a^2*b^2*c^6 - 2*b^4*c^6 - 4*a^2*c^8 + 3*b^2*c^8 - c^10 : :
X(72398) = 5 X[3] - 3 X[16532], 3 X[3] - X[43893], 7 X[3] - 3 X[46451], X[23] - 4 X[62087], 3 X[140] - 2 X[403], 5 X[140] - 4 X[15350], 5 X[140] - 2 X[44267], 3 X[376] + X[35452], 4 X[403] - 3 X[11558], 5 X[403] - 6 X[15350], X[403] - 3 X[34152], 5 X[403] - 3 X[44267], 4 X[468] - 7 X[61784], 5 X[546] - 8 X[5159], 7 X[546] - 8 X[63838], 3 X[547] - 4 X[10257], 7 X[547] - 4 X[47310], 25 X[548] - 4 X[37899], 9 X[548] - 4 X[37931], 7 X[548] - 4 X[47335], 19 X[548] - 4 X[47342], 3 X[549] - X[52403], 5 X[550] + X[5189], 3 X[550] - X[13619], 2 X[858] + X[62151], X[1657] + 3 X[44450], X[2070] - 3 X[8703], 5 X[2071] - X[18403], 3 X[2071] - X[37938], 9 X[2071] - X[64890], 7 X[2072] - 3 X[65087], 5 X[3522] - X[5899], 7 X[3528] - 3 X[37922], 2 X[3530] - 3 X[37948], 3 X[3534] + X[46450], X[3627] - 3 X[65085], 7 X[3853] - 6 X[65087], 3 X[5066] - 2 X[44283], 7 X[5159] - 5 X[63838], 3 X[5189] + 5 X[13619], X[5189] - 5 X[18859], 3 X[7426] - 5 X[15646], 2 X[7426] - 5 X[34200], X[7464] + 2 X[44245], X[7574] + 2 X[62136], 5 X[10096] - 6 X[16532], 3 X[10096] - 2 X[43893], 7 X[10096] - 6 X[46451], 4 X[10151] - 5 X[61940], 7 X[10257] - 3 X[47310], 5 X[11558] - 8 X[15350], X[11558] - 4 X[34152], 5 X[11558] - 4 X[44267], X[11563] - 3 X[37948], 2 X[11799] - 5 X[61790], 3 X[12100] - 2 X[44234], 3 X[12101] - 4 X[23323], X[12103] + 2 X[37950], 5 X[12812] - 2 X[62288], X[13473] - 3 X[15122], 4 X[13473] - 3 X[62026], X[13619] + 3 X[18859], 3 X[14893] - 2 X[64891], 4 X[15122] - X[62026], 2 X[15350] - 5 X[34152], 2 X[15646] - 3 X[34200], 9 X[15688] - X[37949], 3 X[15690] - 2 X[44246], X[15690] + 2 X[54995], 5 X[15712] - 3 X[37943], 9 X[16532] - 5 X[43893], 7 X[16532] - 5 X[46451], 8 X[16976] - 7 X[61821], X[18325] - 4 X[61792], 3 X[18403] - 5 X[37938], 9 X[18403] - 5 X[64890], 2 X[18572] + X[62156], X[20063] - 13 X[62105], 5 X[22248] - 2 X[62344], 3 X[25338] - 4 X[37935], 7 X[25338] - 4 X[47338], 2 X[25338] - 5 X[62064], 5 X[30745] - 2 X[62013], 5 X[34152] - X[44267], X[35001] + 5 X[62104], 3 X[35489] - 7 X[62100], 5 X[37760] - 11 X[62062], 9 X[37899] - 25 X[37931], 7 X[37899] - 25 X[47335], 19 X[37899] - 25 X[47342], 5 X[37923] - 17 X[62084], X[37924] - 7 X[62091], X[37925] - 6 X[62089], 7 X[37931] - 9 X[47335], 19 X[37931] - 9 X[47342], 2 X[37935] - 3 X[37968], 7 X[37935] - 3 X[47338], 8 X[37935] - 15 X[62064], 3 X[37938] - X[64890], 3 X[37941] - 4 X[58190], X[37944] + 4 X[41981], X[37947] - 3 X[44280], 3 X[37955] - 2 X[44264], 3 X[37955] - 5 X[46853], 3 X[37956] - 11 X[62085], 5 X[37958] - 11 X[62079], 7 X[37968] - 2 X[47338], 4 X[37968] - 5 X[62064], 9 X[41982] - 8 X[47114], 3 X[41983] - 2 X[44282], 7 X[43893] - 9 X[46451], 2 X[44214] - 3 X[61782], X[44246] + 3 X[54995], 2 X[44264] - 5 X[46853], 4 X[44452] - 5 X[61810], 8 X[44911] - 9 X[47598], 4 X[44961] - 7 X[61821], 4 X[46031] - 5 X[48154], 4 X[47090] + X[58203], 4 X[47311] + 5 X[62138], 19 X[47335] - 7 X[47342], 4 X[47336] - 7 X[55862], 8 X[47338] - 35 X[62064], 7 X[50693] + X[60466], 5 X[60455] + 7 X[62134], 3 X[60462] + 5 X[62131], 13 X[62092] - X[62290], 3 X[13363] - 2 X[13446]

See Antreas Hatzipolakis and Peter Moses, euclid 9446.

X(72398) lies on these lines: {2, 3}, {74, 50708}, {477, 33639}, {930, 67735}, {1154, 17855}, {1291, 67797}, {1294, 13863}, {2693, 30248}, {2777, 46114}, {6799, 53934}, {13363, 13446}, {13391, 37853}, {13399, 32423}, {13445, 34153}, {14677, 43574}, {22115, 43391}, {29011, 67784}, {40111, 50434}, {53884, 67727}

X(72398) = midpoint of X(i) and X(j) for these {i,j}: {550, 18859}, {3153, 15704}, {13445, 34153}, {14677, 43574}, {16386, 37950}, {40111, 50434}
X(72398) = reflection of X(i) in X(j) for these {i,j}: {140, 34152}, {186, 33923}, {3853, 2072}, {10096, 3}, {11558, 140}, {11563, 3530}, {12103, 16386}, {25338, 37968}, {31726, 3628}, {44267, 15350}, {44961, 16976}, {47096, 22249}
X(72398) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {403, 13473, 44226}, {427, 21284, 65154}, {3520, 13619, 403}, {3530, 50143, 140}, {5159, 7426, 6677}, {5189, 6636, 7426}, {11563, 37948, 3530}, {15690, 66718, 548}, {16387, 47311, 5159}

---

X(72399) = 106TH HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics 4*a^10 - 12*a^8*b^2 + 8*a^6*b^4 + 8*a^4*b^6 - 12*a^2*b^8 + 4*b^10 - 12*a^8*c^2 + 2*a^6*b^2*c^2 - 7*a^4*b^4*c^2 + 29*a^2*b^6*c^2 - 12*b^8*c^2 + 8*a^6*c^4 - 7*a^4*b^2*c^4 - 34*a^2*b^4*c^4 + 8*b^6*c^4 + 8*a^4*c^6 + 29*a^2*b^2*c^6 + 8*b^4*c^6 - 12*a^2*c^8 - 12*b^2*c^8 + 4*c^10 : :
X(72399) = X[2] - 9 X[37943], X[2] - 3 X[44282], 25 X[2] - 9 X[44450], 7 X[2] + 9 X[46451], 17 X[2] - 9 X[65085], 2 X[5] + X[12105], X[23] + 3 X[5055], 7 X[140] + 2 X[47338], 3 X[186] + X[3830], 5 X[381] - X[10296], X[381] + 3 X[37907], 3 X[403] - X[3845], 3 X[403] + X[44265], 4 X[468] - X[18571], 3 X[468] - X[18579], 5 X[468] - 2 X[22249], 2 X[468] + X[44961], 9 X[468] - X[47031], 13 X[468] - X[47308], 11 X[468] + X[47309], 7 X[468] + X[47310], 3 X[468] + X[47332], 5 X[468] - X[47333], 7 X[468] - X[47335], 5 X[468] + X[47336], 3 X[549] - X[54995], 5 X[632] + X[62344], X[858] - 3 X[15699], 5 X[1656] - X[10989], 5 X[1656] + X[37967], 3 X[2070] + 5 X[19709], 3 X[2071] - 7 X[15701], 3 X[2072] + X[47313], 3 X[2072] - 5 X[61910], 7 X[3090] + X[37901], 3 X[3153] - 11 X[61932], 3 X[3524] + X[18325], X[3534] - 3 X[15646], 3 X[3545] - X[18572], 3 X[3545] + 5 X[37760], 2 X[3628] + X[16619], 5 X[3843] + 7 X[37957], 7 X[3851] + 5 X[37953], X[3853] + 2 X[37934], 3 X[5054] - X[37950], X[5066] + 3 X[10096], X[5066] - 6 X[37942], 2 X[5066] - 3 X[46031], 7 X[5066] - 6 X[63838], 11 X[5070] + X[37946], 5 X[5071] - X[7574], 5 X[5071] + 3 X[37909], 2 X[5159] - 3 X[47599], X[5189] - 9 X[61899], 3 X[5899] + 13 X[61901], X[7464] - 5 X[15694], X[7574] + 3 X[37909], 5 X[7575] + X[10296], X[7575] - 3 X[37907], X[8703] + 3 X[11563], 5 X[8703] - 3 X[16386], 2 X[8703] - 3 X[37968], X[8703] - 3 X[44214], X[8703] - 6 X[44900], X[10096] + 2 X[37942], 2 X[10096] + X[46031], 7 X[10096] + 2 X[63838], 2 X[10109] + X[37904], 3 X[10151] - 2 X[61997], 3 X[10257] - 4 X[11540], X[10296] + 15 X[37907], X[10297] + 2 X[44264], X[11001] + 3 X[31726], X[11558] + 2 X[16531], 3 X[11558] + X[62138], 5 X[11563] + X[16386], 2 X[11563] + X[37968], X[11563] + 2 X[44900], 3 X[11799] + X[54995], 2 X[11812] - 3 X[44452], X[12100] - 3 X[44234], 4 X[12811] - X[47339], 3 X[13619] + 5 X[62007], 3 X[14269] + 5 X[37958], 3 X[14892] + 4 X[47316], 2 X[15350] + X[37971], 6 X[15350] - X[47311], 3 X[15350] - 2 X[61896], X[15681] - 5 X[37952], X[15682] - 3 X[44283], X[15685] - 9 X[37955], 5 X[15693] - 3 X[34152], 5 X[15695] - 9 X[37941], 7 X[15703] + X[37924], 5 X[15713] + 3 X[43893], 2 X[16386] - 5 X[37968], X[16386] - 5 X[44214], X[16386] - 10 X[44900], 6 X[16531] - X[62138], 9 X[16532] - X[19710], 3 X[16532] - X[44280], X[18323] - 3 X[23046], 3 X[18403] - 7 X[41106], 3 X[18571] - 4 X[18579], 5 X[18571] - 8 X[22249], X[18571] + 2 X[44961], 9 X[18571] - 4 X[47031], 13 X[18571] - 4 X[47308], 11 X[18571] + 4 X[47309], 7 X[18571] + 4 X[47310], 3 X[18571] + 4 X[47332], 5 X[18571] - 4 X[47333], X[18571] + 4 X[47334], 7 X[18571] - 4 X[47335], 5 X[18571] + 4 X[47336], X[18572] + 5 X[37760], 5 X[18579] - 6 X[22249], 2 X[18579] + 3 X[44961], 3 X[18579] - X[47031], 13 X[18579] - 3 X[47308], 11 X[18579] + 3 X[47309], 7 X[18579] + 3 X[47310], 5 X[18579] - 3 X[47333], X[18579] + 3 X[47334], 7 X[18579] - 3 X[47335], 5 X[18579] + 3 X[47336], 3 X[18859] - 11 X[61843], 5 X[19708] + 3 X[52403], X[19710] - 3 X[44280], X[20063] + 15 X[61906], 5 X[22248] + 3 X[41987], 4 X[22249] + 5 X[44961], 18 X[22249] - 5 X[47031], 26 X[22249] - 5 X[47308], 22 X[22249] + 5 X[47309], 14 X[22249] + 5 X[47310], 6 X[22249] + 5 X[47332], 2 X[22249] + 5 X[47334], 14 X[22249] - 5 X[47335], 2 X[22249] + X[47336], 3 X[23323] - 4 X[61960], X[25338] + 2 X[68319], 5 X[30745] - 9 X[61887], 7 X[33699] - 9 X[65087], X[35001] - 9 X[61864], 4 X[35018] + X[47312], 4 X[35018] - X[47341], 3 X[35452] - 19 X[61857], 9 X[35489] + 7 X[62009], 2 X[37897] + 3 X[47478], X[37899] + 6 X[45757], X[37900] + 9 X[61909], 4 X[37911] - 3 X[47598], 9 X[37922] + 7 X[61974], 5 X[37923] + 11 X[61925], 3 X[37925] + 17 X[61893], 3 X[37931] + 2 X[62010], 6 X[37935] + X[62022], 3 X[37936] + 7 X[61920], 3 X[37938] - X[47314], 3 X[37938] - 7 X[61898], 9 X[37940] + 11 X[61950], 4 X[37942] - X[46031], 7 X[37942] - X[63838], 9 X[37943] + X[44266], 3 X[37943] - X[44282], 25 X[37943] - X[44450], 7 X[37943] + X[46451], 17 X[37943] - X[65085], 3 X[37944] - 23 X[61862], 3 X[37947] + 11 X[61908], 9 X[37948] - 13 X[61797], X[37968] - 4 X[44900], 3 X[37971] + X[47311], 3 X[37971] + 4 X[61896], 3 X[38335] + X[56369], 3 X[44246] - X[62154], X[44266] + 3 X[44282], 25 X[44266] + 9 X[44450], 7 X[44266] - 9 X[46451], 17 X[44266] + 9 X[65085], 25 X[44282] - 3 X[44450], 7 X[44282] + 3 X[46451], 17 X[44282] - 3 X[65085], 7 X[44450] + 25 X[46451], 17 X[44450] - 25 X[65085], 9 X[44961] + 2 X[47031], 13 X[44961] + 2 X[47308], 11 X[44961] - 2 X[47309], 7 X[44961] - 2 X[47310], 3 X[44961] - 2 X[47332], 5 X[44961] + 2 X[47333], 7 X[44961] + 2 X[47335], 5 X[44961] - 2 X[47336], 7 X[46031] - 4 X[63838], 3 X[46450] - 19 X[61913], 17 X[46451] + 7 X[65085], 13 X[47031] - 9 X[47308], 11 X[47031] + 9 X[47309], 7 X[47031] + 9 X[47310], X[47031] + 3 X[47332], 5 X[47031] - 9 X[47333], X[47031] + 9 X[47334], 7 X[47031] - 9 X[47335], 5 X[47031] + 9 X[47336], 3 X[47096] + 7 X[61851], 11 X[47308] + 13 X[47309], 7 X[47308] + 13 X[47310], 3 X[47308] + 13 X[47332], 5 X[47308] - 13 X[47333], X[47308] + 13 X[47334], 7 X[47308] - 13 X[47335], 5 X[47308] + 13 X[47336], 7 X[47309] - 11 X[47310], 3 X[47309] - 11 X[47332], 5 X[47309] + 11 X[47333], X[47309] - 11 X[47334], 7 X[47309] + 11 X[47335], 5 X[47309] - 11 X[47336], 3 X[47310] - 7 X[47332], 5 X[47310] + 7 X[47333], X[47310] - 7 X[47334], 5 X[47310] - 7 X[47336], X[47311] - 4 X[61896], X[47313] + 5 X[61910], X[47314] - 7 X[61898], 5 X[47332] + 3 X[47333], X[47332] - 3 X[47334], 7 X[47332] + 3 X[47335], 5 X[47332] - 3 X[47336], X[47333] + 5 X[47334], 7 X[47333] - 5 X[47335], 7 X[47334] + X[47335], 5 X[47334] - X[47336], 5 X[47335] + 7 X[47336], X[47340] + 4 X[67236], X[47342] + 4 X[61922], 7 X[55856] - X[62332], 3 X[57584] - 5 X[61998], 5 X[60455] - 21 X[61897], 15 X[61882] + X[62290], 7 X[62000] - 3 X[64890], X[62043] - 3 X[64891], X[110] + 3 X[15362], 3 X[5215] - X[38611], X[9158] + 3 X[57305], X[11179] - 5 X[47453], X[11801] + 2 X[15448], 3 X[14643] + X[15360], 2 X[15088] + X[32237], X[20423] + 3 X[47450], X[21850] + 5 X[47452], X[34315] + 3 X[59403], X[34316] + 3 X[59404], 3 X[47455] - X[50979], X[47471] + 3 X[47562], X[50955] + 3 X[52238]

See Antreas Hatzipolakis and Peter Moses, euclid 9446.

X(72399) lies on these lines: {2, 3}, {110, 15362}, {113, 15361}, {524, 10272}, {952, 47495}, {3564, 47544}, {5215, 38611}, {5844, 47488}, {9158, 57305}, {11178, 32217}, {11179, 47453}, {11645, 20304}, {11649, 13364}, {11801, 15448}, {12900, 19924}, {14643, 15360}, {15088, 32237}, {16328, 18487}, {20423, 47450}, {21850, 47452}, {32423, 35266}, {32515, 46986}, {34315, 59403}, {34316, 59404}, {34380, 47473}, {43291, 47169}, {43656, 53950}, {44204, 47219}, {44569, 46817}, {45969, 61606}, {47455, 50979}, {47471, 47562}, {47556, 47581}, {50955, 52238}, {61572, 62508}, {61619, 63124}

X(72399) = midpoint of X(i) and X(j) for these {i,j}: {2, 44266}, {5, 7426}, {113, 15361}, {376, 44267}, {381, 7575}, {468, 47334}, {547, 25338}, {549, 11799}, {3845, 44265}, {10295, 15687}, {10989, 37967}, {11178, 32217}, {11563, 44214}, {11737, 44264}, {15686, 62288}, {16619, 47097}, {18579, 47332}, {44204, 47219}, {44569, 46817}, {47310, 47335}, {47312, 47341}, {47333, 47336}, {47556, 47581}
X(72399) = reflection of X(i) in X(j) for these {i,j}: {547, 68319}, {10297, 11737}, {12105, 7426}, {14893, 37984}, {15122, 10124}, {37968, 44214}, {44214, 44900}, {44961, 47334}, {47097, 3628}, {47333, 22249}, {62139, 66595}
X(72399) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {381, 37907, 7575}, {403, 44265, 3845}, {403, 66725, 37984}, {468, 44961, 18571}, {468, 47332, 18579}, {468, 47336, 22249}, {5071, 37909, 7574}, {10096, 37942, 46031}, {10096, 44233, 25338}, {10109, 66529, 5066}, {10296, 10298, 16386}, {11563, 44900, 37968}, {13626, 13627, 381}, {14002, 37907, 7426}, {18579, 47334, 47332}, {25338, 44234, 25337}, {34330, 62961, 14893}, {44233, 68319, 46031}, {44266, 44282, 2}, {57322, 57323, 61924}

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TWO PROBLEMS CONNECTED

Francisco Javier García Capitán, Two problems [by Antreas Hatzipolakis] connected.

FGJC
geogebra file