A FALSE THEOREM BY VICTOR THEBAULT

Victor Thebault published the following theorem as an exercise:

Si le cercle qui passe par les pieds des bissectrices intérieures d'un triangle est tangent à l'un des côtés, le triangle est isocèle, et réciproquement. 
(If the circle passing through the feet of the interior bisectors of a triangle is tangent to one of the sides, the triangle is isosceles, and vice versa.)

Solution by (A.M.)  [false]
Journal de mathématiques élémentaires.  
75e Annee - No 1 -  1er Octobre 1950, p. 3, #14250

Joseph Andersonn proved that the triangle is not necessarily isosceles.
CERCLE PASSANT PAR LES PIEDS DES BISSECTRICES INTÉRIEURES D'UN TRIANGLE ET TANGENT À L'UN DES CÔTÉS
par A. Monjallon. 
Journal de mathématiques élémentaires.  
75e Annee - No 20, 15 Juillet 1951, pp. 153 - 4 - 

PDF File Victor Thebault

ETC

X(65740) = X(100)X(961)∩X(105)X(5211)

Barycentrics    (a^3 - 2 a^2 b - 2 a b^2 + b^3 + a b c + c^3) (a^3 + b^3 - 2 a^2 c + a b c - 2 a c^2 + c^3) : :

See Antreas Hatzipolakis and Francisco Javier García Capitán, euclid 7092.

X(65740) lies on these lines: {1, 8258}, {28, 2907}, {57, 3882}, {81, 40605}, {88, 3936}, {89, 20090}, {100, 961}, {105, 5211}, {277, 24620}, {278, 3210}, {330, 17740}, {345, 39694}, {646, 30710}, {1022, 4707}, {1054, 17748}, {2006, 37759}, {4612, 64457}, {4850, 39724}, {7052, 37794}, {7132, 37684}, {15474, 17490}, {17282, 39963}, {17495, 21907}, {17776, 39703}, {20882, 37887}, {24183, 30831}, {25430, 56519}, {30699, 65046}, {32779, 39722}, {32849, 39698}, {33116, 56184}, {33168, 35058}, {33655, 37795}, {41839, 56218}

X(65740) = isotomic conjugate of X(37759)

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FJGC A confluence of concepts

Francisco Javier García Capitán,A confluence of concepts
FJGC

FORUM GEOMETRICORUM

FORUM GEOMETRICORUM

All Volumes

FJGC, An unique orthocentroidal system

Euclid 6412

Francisco Javier García Capitán, An unique orthocentroidal system
FJGC

ETC APH-FJGC-EULER

X(44234) = 1ST HATZIPOLAKIS-GARCÍA CAPITÁN-EULER POINT

X(44898) = 2ND HATZIPOLAKIS-GARCÍA CAPITÁN-EULER POINT

X(45306) = 3RD HATZIPOLAKIS-GARCÍA CAPITÁN-EULER POINT

X(45307) = 4TH HATZIPOLAKIS-GARCÍA CAPITÁN-EULER POINT

X(45308) = 5TH HATZIPOLAKIS-GARCÍA CAPITÁN-EULER POINT

X(64480) = 6TH HATZIPOLAKIS-GARCÍA CAPITÁN-EULER POINT

Barycentrics    -a^16 b^2 + 5 a^14 b^4 - 9 a^12 b^6 + 5 a^10 b^8 + 5 a^8 b^10 - 9 a^6 b^12 + 5 a^4 b^14 - a^2 b^16 - a^16 c^2 - 2 a^14 b^2 c^2 + 3 a^12 b^4 c^2 + 13 a^10 b^6 c^2 - 23 a^8 b^8 c^2 + 13 a^6 b^10 c^2 - 6 a^4 b^12 c^2 + 4 a^2 b^14 c^2 - b^16 c^2 + 5 a^14 c^4 + 3 a^12 b^2 c^4 - 28 a^10 b^4 c^4 + 17 a^8 b^6 c^4 + 17 a^6 b^8 c^4 - 7 a^4 b^10 c^4 - 12 a^2 b^12 c^4 + 5 b^14 c^4 - 9 a^12 c^6 + 13 a^10 b^2 c^6 + 17 a^8 b^4 c^6 - 42 a^6 b^6 c^6 + 8 a^4 b^8 c^6 + 28 a^2 b^10 c^6 - 9 b^12 c^6 + 5 a^10 c^8 - 23 a^8 b^2 c^8 + 17 a^6 b^4 c^8 + 8 a^4 b^6 c^8 - 38 a^2 b^8 c^8 + 5 b^10 c^8 + 5 a^8 c^10 + 13 a^6 b^2 c^10 - 7 a^4 b^4 c^10 + 28 a^2 b^6 c^10 + 5 b^8 c^10 - 9 a^6 c^12 - 6 a^4 b^2 c^12 - 12 a^2 b^4 c^12 - 9 b^6 c^12 + 5 a^4 c^14 + 4 a^2 b^2 c^14 + 5 b^4 c^14 - a^2 c^16 - b^2 c^16 + 4 a^13 b c OH S - 6 a^11 b^3 c OH S - 6 a^9 b^5 c OH S + 10 a^7 b^7 c OH S + 6 a^5 b^9 c OH S - 12 a^3 b^11 c OH S + 4 a b^13 c OH S - 6 a^11 b c^3 OH S + 24 a^9 b^3 c^3 OH S - 12 a^7 b^5 c^3 OH S - 24 a^5 b^7 c^3 OH S + 30 a^3 b^9 c^3 OH S - 12 a b^11 c^3 OH S - 6 a^9 b c^5 OH S - 12 a^7 b^3 c^5 OH S + 36 a^5 b^5 c^5 OH S - 18 a^3 b^7 c^5 OH S + 12 a b^9 c^5 OH S + 10 a^7 b c^7 OH S - 24 a^5 b^3 c^7 OH S - 18 a^3 b^5 c^7 OH S - 8 a b^7 c^7 OH S + 6 a^5 b c^9 OH S + 30 a^3 b^3 c^9 OH S + 12 a b^5 c^9 OH S - 12 a^3 b c^11 OH S - 12 a b^3 c^11 OH S + 4 a b c^13 OH S : :

See Antreas Hatzipolakis and Francisco Javier García Capitán, euclid 6388.

X(64480) lies on these lines: {2, 3), {542, 44123}, {1989, 8106}, {8115, 45016}, {13415, 18374}, {15360, 24650}, {32225, 44125}

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X(64481) = 7TH HATZIPOLAKIS-GARCÍA CAPITÁN-EULER POINT

Barycentrics    -a^16 b^2 + 5 a^14 b^4 - 9 a^12 b^6 + 5 a^10 b^8 + 5 a^8 b^10 - 9 a^6 b^12 + 5 a^4 b^14 - a^2 b^16 - a^16 c^2 - 2 a^14 b^2 c^2 + 3 a^12 b^4 c^2 + 13 a^10 b^6 c^2 - 23 a^8 b^8 c^2 + 13 a^6 b^10 c^2 - 6 a^4 b^12 c^2 + 4 a^2 b^14 c^2 - b^16 c^2 + 5 a^14 c^4 + 3 a^12 b^2 c^4 - 28 a^10 b^4 c^4 + 17 a^8 b^6 c^4 + 17 a^6 b^8 c^4 - 7 a^4 b^10 c^4 - 12 a^2 b^12 c^4 + 5 b^14 c^4 - 9 a^12 c^6 + 13 a^10 b^2 c^6 + 17 a^8 b^4 c^6 - 42 a^6 b^6 c^6 + 8 a^4 b^8 c^6 + 28 a^2 b^10 c^6 - 9 b^12 c^6 + 5 a^10 c^8 - 23 a^8 b^2 c^8 + 17 a^6 b^4 c^8 + 8 a^4 b^6 c^8 - 38 a^2 b^8 c^8 + 5 b^10 c^8 + 5 a^8 c^10 + 13 a^6 b^2 c^10 - 7 a^4 b^4 c^10 + 28 a^2 b^6 c^10 + 5 b^8 c^10 - 9 a^6 c^12 - 6 a^4 b^2 c^12 - 12 a^2 b^4 c^12 - 9 b^6 c^12 + 5 a^4 c^14 + 4 a^2 b^2 c^14 + 5 b^4 c^14 - a^2 c^16 - b^2 c^16 - 4 a^13 b c OH S + 6 a^11 b^3 c OH S + 6 a^9 b^5 c OH S - 10 a^7 b^7 c OH S - 6 a^5 b^9 c OH S + 12 a^3 b^11 c OH S - 4 a b^13 c OH S + 6 a^11 b c^3 OH S - 24 a^9 b^3 c^3 OH S + 12 a^7 b^5 c^3 OH S + 24 a^5 b^7 c^3 OH S - 30 a^3 b^9 c^3 OH S + 12 a b^11 c^3 OH S + 6 a^9 b c^5 OH S + 12 a^7 b^3 c^5 OH S - 36 a^5 b^5 c^5 OH S + 18 a^3 b^7 c^5 OH S - 12 a b^9 c^5 OH S - 10 a^7 b c^7 OH S + 24 a^5 b^3 c^7 OH S + 18 a^3 b^5 c^7 OH S + 8 a b^7 c^7 OH S - 6 a^5 b c^9 OH S - 30 a^3 b^3 c^9 OH S - 12 a b^5 c^9 OH S + 12 a^3 b c^11 OH S + 12 a b^3 c^11 OH S - 4 a b c^13 OH S : :

See Antreas Hatzipolakis and Francisco Javier García Capitán, euclid 6388.

X(64481) lies on these lines: {2, 3}, {542, 44124}, {1989, 8105}, {8116, 45016}, {13414, 18374}, {15360, 24651}, {32225, 44126}

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X(64482) = 8TH HATZIPOLAKIS-GARCÍA CAPITÁN-EULER POINT

Barycentrics    -2 a^12 + 6 a^10 b^2 - 13 a^8 b^4 + 5 a^6 b^6 + 7 a^4 b^8 - 5 a^2 b^10 + 2 b^12 + 6 a^10 c^2 - 2 a^8 b^2 c^2 + 11 a^6 b^4 c^2 - 19 a^4 b^6 c^2 + 7 a^2 b^8 c^2 - 9 b^10 c^2 - 13 a^8 c^4 + 11 a^6 b^2 c^4 + 6 a^4 b^4 c^4 + 22 b^8 c^4 + 5 a^6 c^6 - 19 a^4 b^2 c^6 - 30 b^6 c^6 + 7 a^4 c^8 + 7 a^2 b^2 c^8 + 22 b^4 c^8 - 5 a^2 c^10 - 9 b^2 c^10 + 2 c^12 - 2 a^10 W + 5 a^8 b^2 W + 6 a^6 b^4 W - 7 a^4 b^6 W - 4 a^2 b^8 W + 2 b^10 W + 5 a^8 c^2 W - 30 a^6 b^2 c^2 W + 14 a^4 b^4 c^2 W + 25 a^2 b^6 c^2 W - 8 b^8 c^2 W + 6 a^6 c^4 W + 14 a^4 b^2 c^4 W - 46 a^2 b^4 c^4 W + 6 b^6 c^4 W - 7 a^4 c^6 W + 25 a^2 b^2 c^6 W + 6 b^4 c^6 W - 4 a^2 c^8 W - 8 b^2 c^8 W + 2 c^10 W : : where W^2 = a^4 - a^2 b^2 + b^4 - a^2 c^2 - b^2 c^2 + c^4

See Antreas Hatzipolakis and Francisco Javier García Capitán, euclid 6388.

X(64482) lies on these lines: {2, 3}, {2028, 31862}, {3413, 6321}

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X(64483) = 9TH HATZIPOLAKIS-GARCÍA CAPITÁN-EULER POINT

Barycentrics    -2 a^12 + 6 a^10 b^2 - 13 a^8 b^4 + 5 a^6 b^6 + 7 a^4 b^8 - 5 a^2 b^10 + 2 b^12 + 6 a^10 c^2 - 2 a^8 b^2 c^2 + 11 a^6 b^4 c^2 - 19 a^4 b^6 c^2 + 7 a^2 b^8 c^2 - 9 b^10 c^2 - 13 a^8 c^4 + 11 a^6 b^2 c^4 + 6 a^4 b^4 c^4 + 22 b^8 c^4 + 5 a^6 c^6 - 19 a^4 b^2 c^6 - 30 b^6 c^6 + 7 a^4 c^8 + 7 a^2 b^2 c^8 + 22 b^4 c^8 - 5 a^2 c^10 - 9 b^2 c^10 + 2 c^12 + 2 a^10 W - 5 a^8 b^2 W - 6 a^6 b^4 W + 7 a^4 b^6 W + 4 a^2 b^8 W - 2 b^10 W - 5 a^8 c^2 W + 30 a^6 b^2 c^2 W - 14 a^4 b^4 c^2 W - 25 a^2 b^6 c^2 W + 8 b^8 c^2 W - 6 a^6 c^4 W - 14 a^4 b^2 c^4 W + 46 a^2 b^4 c^4 W - 6 b^6 c^4 W + 7 a^4 c^6 W - 25 a^2 b^2 c^6 W - 6 b^4 c^6 W + 4 a^2 c^8 W + 8 b^2 c^8 W - 2 c^10 W : : where W^2 = a^4 - a^2 b^2 + b^4 - a^2 c^2 - b^2 c^2 + c^4

See Antreas Hatzipolakis and Francisco Javier García Capitán, euclid 6388.

X(64483) lies on these lines: {2, 3}, {2029, 31863}, {3414, 6321}

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SAME CENTROID

Let ABC be a triangle and A'B'C' the cevian triangle of O

Denote

Ma, Mb, Mc = the midpoints of AA'. BB', CC', resp.

Ha, Hb, Hc = the orthocenters of OMbMc, OMcMa, OMaMb, resp,

The triangles ABC and HaHbHc share the same centroid G

APH

Francisco Javier García Capitán A triangle of orthocenters with centroid G
FJGC