A
Adams circle
Let ABC be a triangle with A'B'C'. Through the X(7) of ABC draw parallel lines to B'C', C'A', A'B'. These lines intersect the sidelines of ABC in six concylic points lying on the . ( See Index of triangles referenced in ETC ).
Ajima-Malfatti points
X(179) and X(180): In triangle ABC, if A' is the touchpoint of the B- and C- inner/outer and B', C' are defined cyclically, then the lines AA', BB', CC' concur at X(179)/X(180).
altitude (in a triangle)
Line through a vertex of the triangle and perpendicular to the opposite side.
angle of similarity or similitude
See .
anticenter
The point of concurrence of the four of a cyclic quadrilateral.
Source: Wolfram's Anticenter.
anticevian line
In a triangle ABC, let A'B'C' be the of a point P. Denote A"=(BC)∩(B'C'). The line AA" is the A-anticevian line of P.
anticevian triangle
Given a point P, the anticevian triangle A'B'C' of a triangle ABC with respect to P is a triangle such that:
1. B'C' passes through A, C'A' passes through B, and A'B' passes through C;
2. AA', BB', and CC' pass through P;
3. ABC is the of A'B'C' with respect to P.
Source: Wolfram's Anticevian Triangle.
antigonal conjugate
Let ABC be a triangle, P a point and A', B', C' the reflections of P in BC, CA, AB, respectively. Then circles {{B, C, A'}}, {{C, A, B'}}, {{A, B, C'}} concur in the antigonal conjugate of de P.
antiorthic axis
X(44)X(513): of the . ( See Index of triangles referenced in ETC ).
antiparallel lines
Let 𝓁1 and 𝓁2 be two distinct lines.Two other lines 𝓂1 and 𝓂2 (none of them perpendicular to 𝓁1 or 𝓁2) are antiparallel with respect to 𝓁1 and 𝓁2 if the acute (or obtuse) angle between 𝓂1 and 𝓁1 has the same measure than the acute (or obtuse) angle between 𝓂2 and 𝓁2.
Notes:
1) The line joining the feet to two of a triangle is antiparallel to the third side.
2) The tangent to a triangle's circumcircle at a vertex is antiparallel to the opposite side.
3) The radius of the circumcircle at a vertex is perpendicular to all lines antiparallel to the opposite sides.
antipedal line
Let ABC be a triangle, P a point on its and P* the of P. The line PP* is the antipedal line of P.
antipedal triangle
Given a triangle ABC and a point P, the antipedal triangle of P with respect to ABC is the triangle A'B'C' bounded by the perpendicular lines through A, B, C to AP, BP, CP, respectively.
antisymmetric triangle
Let ABC be a triangle and P a point. Let A', B', C' be the reflections of A, B, C in P, respectively. A'B'C' is the antisymmetric triangle of ABC in P.
antitomic conjugate
The of the inverse in the (= of the line-at-infinity) of the isotomic conjugate of P.
Source: Preamble just before X(14941).
apollonian circles
In a triangle ABC, the internal and external bisectors of angle A cut BC at Ai and Ae, respectively. The circle with diameter AiAe is the A-apollonian circle of ABC.
Note: The A-apollonian circle of ABC passes through A and both X(15) and X(16).
Apollonius circle
Circle internally tangent to the three excircles.
Apollonius point
X(181): In a triangle ABC, let A' be the touchpoint of the A-excircle and the and define B', C' cyclically. Lines AA', BB', CC' concur at X(181).
areal center
Let A'B'C', A"B"C" be two triangles inscribed in ABC. The areal center of both inscribed triangles is the point Q such that triangles QA'A", QB'B", QC'C" have the same areas.
areal coordinates
Let ABC be a triangle and P a point. Let X be the area of triangle PBC, considering X positive if triangles ABC and PBC have the same orientation, or negative otherwise. Define Y, Z similarly, by taking triangles PCA and PAB, respectively. The ordered set {X, Y, Z} are the actual areal coordinates of P. P is univocally determined with these coordinates.
Proportional values to X, Y, Z can be instead for univocally determining P, i.e., x = k*X, y = k*Y, z = k*Z, where k≠0 is a constant. In this case x : y : z are the areal coordinates of P.
It is most common to express x : y : z as a normalized triad, i.e. , such that x + y + z = 1.
Note: The name barycentric coordinates is used universally instead of areal coordinates.
Artzt parabolas
In a triangle ABC, the A-Artzt parabola is the parabola tangent to AB and AC at B and C, respectively.
auto-polar triangle
See .
axis of perspective
See .
axis of similarity or similitude
See .
B
B-Line
A line that simultaneously bisects a triangle's perimeter and area.
Source: Wolfram's B-Line.
bare angle center
X(1049): Center with trilinear coordinates A : B : C.
barycenter
See .
barycentric coordinates
See .
BCI triangle
Let I be the incenter X(1) of ABC and Ia, Ib, Ic the incenters of BCI, CAI, ABI, respectively. IaIbIc is the BCI triangle of ABC.
Begonia point
Let ABC be a triangle and A'B'C' the cevian triangle of a point P. If A", B", C" are the reflections of P in B'C', C'A' and A'B', respectively, then the lines AA", BB" CC" concur at the Begonian point of P.
Source: Wolfram demonstrations.
Beltrami points
Inverse of in the circumcircle.
Source: TTW - The Triangle Web.
Bevan circle
See .
Bevan point
X(40): Center of the .
Bevan-Schröder point
X(1319): Inverse of X(56) in the circumcircle.
Source: TTW - The Triangle Web.
bicentric pair
See .
Bickart points
X(39158) and X(39159): Real foci of the .
bilogic triangles
If two triangles are and , it is said that they are bilogic triangles.
Source: Anopolis 1880.
binary cofactor triangle
Let T1 = A1B1C1 and T2 = A2B2C2 two triangles with A1≠A2, B1≠B2 and C1≠C2. The binary cofactor triangle of T1 and T2, denoted 𝔹(T1, T2), is the triangle T* = A*B*C*, where A* is the of A1 ⨯ A2, B* is the of B1 ⨯ B2 and C* is the of C1 ⨯ C2.
Brianchon point
of an inconic of a triangle.
Brianchon theorem
« If a hexagon is circumscribed to a (maybe degenerate) conic then the lines joining the opposite vertices are concurrent, and reciprocally. »
Brocard 1st circle
Circle with diameter X(3)X(6)
Brocard 1st parabola
The A-1st Brocard parabola of a triangle ABC has focus the A-vertex of the and directrix the A- of ABC. ( See Index of triangles referenced in ETC ).
Source: TTW - The Triangle Web.
Brocard 2nd circle
Circle centered at the X(3) and passing through both .
Brocard 2nd parabola
The A-2nd Brocard parabola of a triangle ABC has focus the A-vertex of the and directrix the A- of ABC.
Source: TTW - The Triangle Web.
Brocard 3rd point
X(76) = of X(6)
Brocard angle
The 1st Brocard point W1 of a triangle ABC is the interior point of ABC for which ∡W1AB = ∡W1BC = ∡W1CA = ω1.
The 2nd Brocard point W2 of a triangle ABC is the interior point of ABC for which ∡W2AC = ∡W2BA = ∡W2CB = = ω2.
It results that ω1 = ω2 = ω, the Brocard angle of ABC.
Brocard axis
X(3)X(6): Line joining the and the .
Brocard diameter
Segment joining the X(3) and the X(6).
Brocard inellipse
Inconic of a triangle with center on its .
Brocard midpoint
X(39): Midpoint of .
Brocard points
See .
C
center function
See .
center of (direct/inverse) similitude
See .
center of perspective
See .
centroid (of a triangle)
X(2): Point of intersection of the three . Also named barycenter.
Ceva-conjugate
P- Ceva-conjugate-of-U = of the of P and the of U.
Ceva-point
Let A'B'C' be the of U. Let A" = PA' ∩ BC, and define B" and C" cyclically. The cevapoint of P and U is the of triangles ABC and A"B"C".
cevian circle
Circumcircle of a .
cevian line
See .
cevian nest
Let T1, T2, T3 be three triangles such that T3 is inscribed in T2 and T2 is inscribed in T1. This set of triangles is called a cevian nest.
Note: In a cevian nest, if a pair of triangles are then each is perspective to the third.
cevian product
See .
cevian trace
See .
cevian triangle
Let ABC be and a triangle a P a point distinct of its vertices. Denote A' = AP∩BC, B' = BP∩CA, C'=CP∩AB. Then A'B'C' is the cevian triangle of P.
The points A', B', C' are the cevian traces of P and the lines AA', BB', CC' are the cevian lines of P.
Chasles polar triangle theorem
«A triangle and its with respect to a given conic are always .»
Note: The of both triangles is called the conic perspector.
circle of similitude
If {a}, {b} are two circles with E and I, then the circle with diameter EI is the circle of similitude of {a} and {b}.
circle power
Let Ω be a circle and P a point. Let T', T" be the touchpoints of Ω and its tangents through P. The circle power of P with respect to Ω is the squared distances PT'2 = PT"2.
circlecevian triangle
Let P be a point and A' the intersection, other than P, of the line AP and the circle PBC. Define B', C' cyclically. A'B'C' is the circlecevian triangle of P
circumanticevian triangle
Let ABC be a triangle, P a point and A'B'C' the of P. Let A" be the intersection, other than A, of B'C' and the circumcircle of ABC, and build B", C" cyclically, Then A"B"C" is the circumanticevian triangle of P.
circumantipedal triangle
Let ABC be a triangle, P a point and A'B'C' the of P. Let A" be the intersection, other than A, of B'C' and the circumcircle of ABC, and build B", C" cyclically, Then A"B"C" is the circumantipedal triangle of P.
circumcenter
X(3): Center of the of a triangle.
circumcevian triangle
Let ABC be a triangle and P a point. Let A' be the point, other than A, where line AP cuts again the circumcircle of ABC; define B', C' similarly. A'B'C' is the circumcevian triangle of P.
circumcircle
Circle through the vertices of a triangle.
circumconic
Conic passing through the three vertices of a triangle.
Clawson point
X(19): The is homothetic to the , and its is known as the Clawson point, or sometimes the crucial point.
( See Index of triangles referenced in ETC ).
cocevian triangle
Let ABC be a triangle, P a point not on its sidelines, and A'B'C' the of the of P. The cocevian triangle of P is the of ABC and A'B'C'
( 𝔹(ABC, A'B'C') ).
congruent isoscelizers point
X(173): There is a unique configuration of for a given triangle such that all three have the same length and are concurrent. X(173) is such point of concurrence.
congruent squares point
See .
conic perspector
See: .
Conway circle
Let ABC be a triangle.
Theses six built points lie on a circle named the Conway circle.
- On the opposite ray of ray AB, build the point Ab such that AAb = BC.
- On the opposite ray of ray AC, build the point Ac such that AAc = BC.
- Build Bc, Ca and Ba, Cb cyclically, as in 1. and 2.
Conway point
X(384): Intersection of and line X(32)X(76).
cosine circle
In a triangle ABC, draw through the X(6). The points where these lines intersect the triangle sidelines lie on the cosine circle (sometimes called the ).
cross conjugate
Let ABC be a triangle, P a point and A'B'C' be the of another point U. Let A"= PA' ∩ B'C', and define B" and C" cyclically. Triangles ABC and A"B"C" are and their is the P-cross conjugate of U.
cross product
Let P1 = u1 : v1 : w1 and P2 = u2 : v2 : w2. The cross product of P1 and P2, denoted P1⨯P2, is the point P* = v1*w2 - w1*v2 : w1*u2 - u1*w2 : u1*v2 - v1*u2.
cross-cevian point
See .
cross-cevian triangle
Let T=ABC be a triangle, P', P" two distinct points and and T'=A'B'C', T"=A"B"C" the of P', P", respectively, with respect to T. Define A* = AA" ∩ B'C', B* = BB"∩ C'A', C* = CC"∩A'B'. Triangle A*B*C* is the cross-cevian triangle of P' and P" or the cross-cevian triangle of T' and T" and denoted as ℂ(P', P") or ℂ(T', T").
ℂ(T', T") is perspective to T' and the perspector, denoted 𝒞(P', P"), is named the cross-cevian point of P' and P".
cross-triangle
Let A'B'C' and A"B"C" be two triangles, neither inscribed in the other. The cross-triangle of A'B'C' and A"B"C" is the triangle A*B*C* with A* = B'C"∩B"C', B* = C'A"∩C"A', C*=A'B"∩A"B'.
crucial point
See .
cubic
A degree-3 curve. For a site devoted to cubics and other curves with higher degrees, visit Bernard Gibert's "Cubics in the triangle plane".
Cundy-Parry Phi (Φ) transformation
Let ABC be a triangle with circumecenter O and orthocenter H, and let P be a point whose isogonal with respect to ABC is P*. The Cundy-Parry Φ transformation of P is the point Q(P) given by Q(P)=OP ∩ HP*.
Cundy-Parry Psi (Ψ) transformation
Let ABC be a triangle with circumecenter O and orthocenter H, and let P be a point whose isogonal with respect to ABC is P*. The Cundy-Parry Ψ transformation of P is the point Q(P) given by Q(P)=HP ∩ OP*.
cyclocevian conjugate
Let ABC be a triangle, P a point not on its sidelines and A'B'C' the of ABC. Let A", B", C" be the points of intersection, others than A', B', C', at which the circumcircle of A'B'C' cuts BC, CA, AB, respectively. Then the lines AA", BB", CC" concur in the cyclocevian conjugate of P.
cyclologic center
See .
cyclologic triangles
Let T' = A'B'C', T" = A"B"C" be two triangles. Draw the circles {{a'}} = A'B"C", {{b'}} = B'C"A", and {{c'}}= C'A"B" and draw the circles {{a"}} = A"B'C', {{b"}} = B"C'A' and {{c"}} = C"A'B'. It has been proved that if circles {{a'}}, {{b'}}, {{c'}} concur in a point P' then the circles {{a"}}, {{b"}}, (c") concur in a point P", and reciprocally. In this case, T' and T" are said to be cyclologic triangles and P', P" are called the cyclologic centers T' to T" and T" to T', respectively.
cyclotomic conjugate
See .
D
De Longchamps circle
of .
Note: Only for obtuse triangles.
de Longchamps ellipse
Ellipse circumscribing the of X(57) and the . ( See Index of triangles referenced in ETC ).
de Longchamps line
Perpendicular to through X(858).
De Longchamps point
X(20): Reflection of the X(4) in the X(3).
de Villiers points
X(1127): of ABC and . ( See Index of triangles referenced in ETC ).
X(1128): of ABC and . ( See Index of triangles referenced in ETC ).
Desargues theorem
« Two triangles are in axial-perspective if, and only if, they are in central-perspective. »
Descartes formula for a triad of pairwise tangent circles
Let three circles 𝒸i with radii ri (i=1,2,3) be tangent by pairs. Let ki = 1/ri be the curvature of 𝒸i. Then the curvatures Ki, Ke of the circles internally and externally tangent to 𝒸1, 𝒸2 and 𝒸3 satisfy:
K = k1 + k2 + k3 ± Sqrt( k2*k3+k3*k1+k1*k2)
desmic product
See .
Dou circle
The Dou circle of a triangle ABC is the circle cutting its sidelines in A',A", B', B", C', C" and such that ∡ A'AA" = ∡ B'BB" = ∡ C'CC" = π/2.
Dou circle has center X(155) and squared-radius (4*R^6-(5*R^2-SW)*S^2)/(4*R^2-SW)^2.
Droussent point
X(316): Anticomplement of the X(187).
Source: TTW - The Triangle Web.
Droz-Farny circles
Let AmBmCm be the medial triangle of ABC. The circle with center A and passing through the circumcenter X(3) of ABC cuts BC in A', A". Build B', B", C', C" cyclically. Then these six points lie on a circle named the 1st Droz-Farny circle.
Let AmBmCm be the medial triangle of ABC. The circle with center A' and passing through the orthocenter X(4) of ABC cuts BmCm in A', A". Build B', B", C', C" cyclically. Then these six points lie on a circle named the 2nd Droz-Farny circle.
duality of a point and a line
A point P and a line ℓ are dual with respect to a triangle if P is the of the of ℓ, or, equivalently, if ℓ is the of the of P.
E
Ehrmann congruent squares point
X(1144): Consider a point P inside a reference triangle ABC, construct line segments AP, BP, and CP. The Ehrmann congruent squares point is the unique point P such that three equal squares can be inscribed internally on the sides of ABC such that they touch the line segments in exactly two points each. Also called congruent squares point.
Source: Wolfram's Ehrmann congruent point.
eigencenter
Let T be a central triangle and 𝕌(T) its . Then T and 𝕌(T) are , and their is called the eigencenter of T.
Eppstein points
X(481): Let A'B'C' be the , and A" be the contact point of the B- and C- ; B" and C" are defined cyclically. Then the lines A'A", B'B", C'C" concur at the 1st Eppstein point.
X(482): Let A'B'C' be the , and A" be the contact point of the B- and C- ; B" and C" are defined cyclically. Then the lines A'A", B'B", C'C" concur at the 2nd Eppstein point.
( See Index of triangles referenced in ETC ).
equal detour point
X(176): A point X is defined as a point of equal detour of a triangle ABC if |XB| + |XC| - |BC| = |XC| + |XA| - |CA| = |XA| + |XB| - |AB|.
Notes:
- This point is also the center of the .
- The condition tan(A/2) + tan(B/2) + tan(C/2) = 2 is necessary and sufficient for the existence of exactly one point of equal detour.
equal parallelians point
X(192): Let ABC be a triangle and P a point.
Through P:
1) draw the parallel line to BC cutting AB, AC in Ac and Ab,
2) draw the parallel line to CA cutting BC, BA in Ba and Bc,
3) draw the parallel line to AB cutting CA, CB in Cb and Ca.
Then, for P=X(192), AcAb = BaBc = CbCa and this point is called the equal parallelians point.
equi-Brocard center
X(368): The only point P such that triangles BCP, CAP, ABP have the same .
equicenter
Let T' = A'B'C' and T" = A"B"C" be two triangles or, in general, two geometric figures. The point P having the same relative position with respect to both triangles (or figures) is the equicenter of T' and T".
Note: Algebraically, the equicenter of T' and T" is the fixed point of the transformation {A', B', C'} → {A", B", C"}.
equilateral cevian triangle point
X(370): The only point whose is equilateral.
equilogic triangles
Two triangles A'B'C' and A"B"C" are equilogic if A'A" = B'B"=C'C".
Euler circle
See .
Euler infinity point
X(30): Intersection of the Euler line and line at infinity.
Euler line
The line joining the circumcenter X(3) and the othocenter X(4) of a triangle.
Euler reflection point
X(399): See Forum Geometricorum Volume 10 (2010) 157-163.
Euler-Gergonne-Soddy circle
Circumcircle of . Also known as EGS circle. ( See Index of triangles referenced in ETC ).
eulerologic center
See .
eulerologic triangles
Two triangles T' = A'B'C' y T" = A"B"C" are eulerolgic if the Euler lines of A'B"C", B'C"A" y C'A"B" are cocurrent. In this case, such point of concurrence is the eulerologic center T' to T".
Note: The existence of the does not imply the existence of the
Evans conic
Conic through , and .
Evans perspector
X(484): Inverse of X(35) in the circumcircle.
Source: TTW - The Triangle Web.
Evans point
X(1375): Intersection of and .
excentral circle
Circle through the excenters, also called Bevan circle.
excentral-hexyl ellipse
The excentral-hexyl ellipse is the ellipse passing through vertices of the and . ( See Index of triangles referenced in ETC ).
excosine circles
In a triangle ABC, let A' be the pole of the sideline BC with respect to the circumcircle of ABC. The circle with center A' and passing through B passes also through C. This circle is the A-excosine circle.
Exeter point
X(22): of the of the X(2) and the . ( See Index of triangles referenced in ETC ).
exsimilcenter
See .
extangent (line)
Given a triangle ABC, there are four lines simultaneously tangent to the B- and C-excircles. Of these, three correspond to the sidelines of the triangle, and the fourth is known as the A-extangent.
Source: Wolfram's Extangent.
extangents circle
Circumcircle of . ( See Index of triangles referenced in ETC ).
extriangle triangle
Given a triangle ABC and a point P, the extriangle triangle of P is the of the of P.
F
far-out point
X(23): Inverse in circumcircle of X(2)
Fermat axis
X(13)X(14) = X(6)X(13): Line through the .
Fermat points
X(13) and X(14). Also called isogonic centers.
Feuerbach circle
See .
Feuerbach hyperbola
Circumhyperbola of a triangle centered at the X(5).
Feuerbach point
X(11): Touchpoint of the incircle and the .
Fletcher point
X(1323): The intersection of the and the .
Fontené theorems
«Let ABC be a triangle, Tm = AmBmCm the medial triangle, P a point and Tp = ApBpCp the pedal triangle of P. Let A* = BmCm ∩ BpCp and similarly B*, C*. Then the lines A*Ap, B*Bp, C*Cp concur in a point on the circumcircles of Tm and Tp.»
Source: Wolfram's Fontené theorems.
Fregier point and Fregier theorem
«Pick any point P on a conic section, and draw a series of right angles having this point as their vertices. Then the line segments connecting the rays of the right angles where they intersect the conic section concur in a point P'.»
Source: Wolfram's Frégier's Theorem.
Fuhrmann circle
Circumcircle of the . ( See Index of triangles referenced in ETC ).
Fuhrmann point
X(355): Midpoint of the X(8) and the X(4).
G
Gallatly circle
In a triangle, the Gallatly circle is the circle with center at the X(39) and radius R*sin(ω), where ω is the of the triangle.
gamma triangle
Let T=ABC be a triangle and Ai the point at which the perpendicular line through A to BC cuts the line at infinity. Let A' be the of Ai, on the circumcircle of ABC, and define B', C' cyclically. Triangle A'B'C' is the gamma triangle of T. From here, it can be affirmed that all triangles homothetic to T have the same gamma triangle.
GEOS circle
In a triangle, let (G) be the , (E) the, (O) the and (S) the . Then (G)∩(S), (E)∩(O), (O)∩(G) and (S)∩(E) lie on the GEOS circle.
Gergonne line
Line X(241)X(514): The Gergonne line is the of a triangle and its . ( See Index of triangles referenced in ETC ).
Gergonne point
X(7): In a triangle ABC, let A'', B'', C'' be the touchpoints of the incircle and sidelines of ABC. Lines (AA''), (BB'') y (CC'') concur at Georgonne point.
Gibert point
X(1141): Perspector of ABC and the of X(1157)
Gob point
X(25): of the and the . ( See Index of triangles referenced in ETC )
Source: TTW - The Triangle Web
Gossard point
X(402): of ABC and the . ( See Index of triangles referenced in ETC )
Source: TTW - The Triangle Web
Grebe point
X(6): Point of intersection of the three .
Griffiths points
X(1373): of the ( See Index of triangles referenced in ETC ) and its .
X(1374): of the ( See Index of triangles referenced in ETC ) and its .
Griffiths theorems
1) «If a point P moves along a line r through the of a given triangle T, then the of P with respect to T passes through a fixed point (the Griffiths point) on the of T». This point is the of r.
2) «If A, B, C, D are four points on a circle and r is a line through the center of the circle, then the Griffith points of r with respect to triangles BCD, CDA, DAB and ABC are collinear».
H
Haimov triangle
Let A'B'C' the of P in ABC. Circles (ABB') and (ACC') cuts again in A". Define B" and C" cyclically. The triangle A"B"C" is the Haimov triangle of P.
half-altitude circle
See .
half-Moses circle
Circle with center X(39) and radius R*sin(ω)^2, where ω is the of the triangle.
hexyl circle
Circumcircle of . ( See Index of triangles referenced in ETC ).
Hofstadter point
See .
Hofstadter points
X(359) & X(360)
Hofstadter triangle
In a triangle ABC, let:
Build Bk and Ck cyclically.
- k: a real number;
- (ab): rotation with angle +k*∡B of line (BC) around vertex B;
- (ac): rotation with angle -k*∡C of line (BC) around vertex C;
- Ak = (ab) ∩ (ac)
AkBkCk is the k-Hofstadter triangle of ABC.
Note: For any k, the k-Hofstadter triangle is to ABC and the perspector is named the k-Hofstadter point.
Source: Hofstadter points (Wikipedia).
homologous points
See .
homology center
See .
homothetic center
See .
homothetic triangles
See .
I
incenter
X(1): Center of the .
incenter-excenter circles
Let ABC be a triangle with incenter X(1) = I and A-excenter = Ia. The circle with diameter IIa is the A-excenter-incenter circle of ABC.
incentral circle
The incentral circle is the circumcircle of the . ( See Index of triangles referenced in ETC )
incircle
Circle inscribed in a triangle and tangent to its three side-segments.
inconic
Conic inscribed in a triangle.
insimilcenter
See .
intangent (line)
Given a triangle ABC, there are four lines simultaneously tangent to the and the A-. Of these, three correspond to the sidelines of the triangle, and the fourth is known as the A-intangent.
Source: Wolfram's Intangent.
intangents circle
Circumcircle of . ( See Index of triangles referenced in ETC )
intriangle triangle
Given a triangle ABC and a point P, the intriangle triangle of P is the of the of P.
isodynamic points
X(15) and X(16)
isogonal conjugate
Let ABC be a triangle and P a point. Reflect line AP in the internal bisector of angle A, line BP in the internal bisector of angle B, and, line CP in the internal bisector of angle C. Then these three lines concur in the isogonal conjugate of P. It is also called triangular inverse of P.
isogonal line
The reflection of a line through a vertex of a triangle in the internal angle bisector of the angle with the same vertex.
isogonic centers
See .
isologic points
X(5002) and X(5003): Let ABC be an acute triangle and ωa its A-. Let Ωa the reflection of ωa in the perpendicular bisector of BC. Build Ωb and Ωc similarly. Then Ωa , Ωb, Ωc are coaxial and intersect in the isologic points.
Source: TTW - The Triangle Web.
isoperimetric point
X(175): Let ABC be a triangle with BC=a, CA=b, AB=c, R=radius of the circumcircle and r=radius of the incircle and such that a + b + c > 4*R + r. Then S=X(175) is the unique point such that triangles SBC, SCA, SAB have equal perimeters and it is called the isoperimetric point.
Note: This point is also the center of the .
isoscelizer
In a triangle ABC, an A-isoscelizer is a segment B'C', with B', C' lying on AC, AB, respectively, such that AB' = AC', i.e., such that triangle B'AC' is isosceles.
isotomic conjugate
Let ABC be a triangle, Am, Bm, Cm the midpoints of BC, CA, AB, respectively, P a point and A'B'C' the cevian triangle of P.
Let:
1) A" = reflection of A' in Am,
2) B" = reflection of B' in Bm,
3) C" = reflection of C' in Cm.
Then the lines AA", BB", CC" concur in the isotomic conjugate of P.
isotomic line
Let ABC be a triangle, Am the midpoint of BC and a line through A cutting BC in A'. The isotomic line of AA' is the line AA", where A" is the reflection of A' in Am.
J
Jacobi point
See .
Jacobi triangles
Given a triangle ABC, build three triangles Ta = A'CB, Tb = B'AC, Tc = C'BA, all outwards ABC (or all inwards ABC), and such that, Tb and Tc have angles in A with the same measure α, Tc and Ta have angles in B with the same measure β, and Ta and Tb have angles in C with the same measure γ. These triangles are the Jacobi triangles of (α, β, γ).
Notes: In each case, Jacobi triangles built outwardly/inwardly ABC and for any (α, β, γ), they are similar triangles (see ). and the lines AA', BB', CC' concur at the external/internal Jacobi point of (α, β, γ) .
Jenkins circles
The Jenkins circles are the three circles internally tangent to an excircle and externally tangent to the other two.
Source: TTW - The Triangle Web.
Jerabek center
X(125): Center of the .
Jerabek hyperbola
Circumhyperbola through the X(3) and the X(4) of a triangle.
Johnson circles
In a triangle ABC with orthocenter H, the circles {{B, C, H}}, {{C,A,H}}, {{A,B,H}} are the Johnson circles of ABC.
Note: By , all these circles have the same radius than the circumcircle of ABC.
Johnson circumconic
Circumconic of a triangle passing also through the vertices of its . ( See Index of triangles referenced in ETC ).
Johnson theorem
«If three given congruent circles intersect all in a point P and, by pairs, in A, B, C, then the circle {{A, B, C}} is congruent to the given circles.»
Johnson-Yff circles
The are the two triplets of congruent circles in which each circle is tangent to two sides of a reference triangle.
Since each triplet of are congruent and pass through a single point, they obey . As a result, in each case, there is a fourth circle congruent to the original three and passing through the points of pairwise intersections. These last circles are the Johnson-Yff circles.
K
Kenmotu circles
See .
Kenmotu points
X(371): 1st Kenmotu point.
X(372): 2nd Kenmotu point.
Inscribe three equal squares in a triangle such that each square touches two sides and all three squares touch at a single common point. This common point is the 1st Kenmotu point.
The contact points of the squares with the sides are concyclic and lie on the 1st Kenmotu circle.
There is a second version of the above construction for which the equal squares have a common vertex ( 2nd Kenmotu point and each square touches two sides of the triangle, although the squares are not inscribed in the triangle. The points of contact on the sidelines of the triangle are also concyclic and lie on the 2nd Kenmotu circle.
See here.
Kiepert center
X(115): Center of the .
Kiepert hyperbola
In a triangle ABC, build isosceles triangles BCA' , CAB', ABC' (all inwards or all outwards ABC), with bases BC, CA, AB, respectively, and all having angles in the base equal to Θ. Then the lines AA', BB', CC' concur in a point P(θ), i.e., ABC and A'B'C' are , and the locus of P(θ) when θ varies is a circumhyperbola of ABC, named the Kiepert hyperbola.
Kiepert parabola
In a triangle ABC, build isosceles triangles BCA' , CAB', ABC' (all inwards or all outwards ABC), with bases BC, CA, AB, respectively and all having angles in the base equal to Θ. Then the lines AA', BB', CC' concur in a point P(θ), i.e., ABC and A'B'C' are perspective triangles, and the envolvent of the when Θ varies is a parabola, named the Kiepert parabola.
Kosnita point
X(54): of X(5)
L
Lemoine 1st circle
Draw lines through the (X(6)) and parallel to the sides of the triangle ABC. The points where these parallel lines intersect the sides of ABC lie on a circle known as the 1st Lemoine circle or, sometimes, as the triplicate-ratio circle .
Lemoine 2nd circle
See .
Lemoine 3rd circle
of . ( See Index of triangles referenced in ETC )
Lemoine axis
Line X(187)X(237); of a triangle and its .
Lemoine inellipse
Inconic with center X(597).
Lemoine point
X(6): Point of intersection of the three .
Lester circle
Circle through the X(3), the X(5) and both X(13) and X(14).
line at infinity
X(30)X(511)
line-polar triangle
Let ABC be a triangle and A', B', C' three non collinear points. The line-polar triangle of A', B', C' is the triangle with vertices in the of the lines B'C', C'A', A'B'.
Longuet-Higgins circle
In a given triangle ABC, let BC= a. CA= b, AB = c. Build the circles centered at A, B, C with radii b+c, c+a, a+b, respectively. The radical circle of these circles is the Longuet-Higgins circle. The center of this circles is X(962), called the Longuet-Higgins point.
Longuet-Higgins point
X(962): See .
Lucas central circles
Circumcircles of . ( See Index of triangles referenced in ETC )
Lucas circles
Inscribe a square A'A"AbAc in a triangle ABC such that A', A" lie on BC, Ab lies on CA and Ac lies on AB. This square is the A-inner-Lucas square and the circumcircle of AAbAc is the A-inner-Lucas circle. Similar constructions lead to B- and C- squares and circles.
A second A-square can be built, with A' and A" on BC, and the other vertices of the square each on the other sides of ABC, although the square is not inscribed in ABC. Therefore, there exist an A-outer-Lucas square and an A-outer-Lucas circle, and B- and C- versions can be built cyclically. See a figure here.
Lucas inner circle
The circle externally tangent to the three inner-.
Note: The circle internally tangent to the three inner- is the circumcircle.
Lucas squares
See .
M
MacBeath circle
Circumcircle of the . ( See Index of triangles referenced in ETC )
MacBeath circumconic
Circumconic with center in the X(6).
MacBeath inconic
Inconic with center in the X(5).
major triangle center
A u : v : w is called a major triangle center if the triangle center function u=f(a,b,c,A,B,C) is a function of angle A alone, and therefore beta and gamma of B and C alone, respectively.
Source: Wolfram's Major Triangle Center.
Malfatti circles
Three circles in the interior of a triangle, each one tangent to the other two and to two sides of the triangles. These circles are better named inner-Malfatti circles.
There is a second triad of circles each tangent to two sidelines of ABC and to the other two circles, but not packed inside ABC. These circles are the outer-Malfatti circles.
maltitude
A maltitude ("midpoint altitude") is a perpendicular drawn to a side of a quadrilateral from the midpoint of the opposite side.
Source: Wolfram's maltitudes.
Mandart circle
Circumcircle of the . ( See Index of triangles referenced in ETC )
Mandart inellipse
Inellipse centered at the X(9).
Mannheim circles
See .
McCay circles
Let G be the X(2) of a triangle ABC and A'B'C' the of ABC ( See Index of triangles referenced in ETC ). Circles {(GB'C'}), {(GC'A'}) and ({GA'B'}) are A-, B-, C- McCay circles, respectively.
median (of a triangle)
Line joining a vertex of the triangle and the midpoint of the opposite vertex.
mediatrix (pl. mediatrices)
Perpendicular bisector of a segment.
Sources: Wikipedia, RBJLabs
meta-triangle
Let T' = A'B'C' and T" = A"B"C" be two non-. Let A* = B'B"∩C'C", B* = C'C"∩A'A", C* = A'A"∩B'B". A*B*C* is the meta-triangle of T' and T".
mid-arc points
In a triangle ABC, the A-mid-arc point is the midpoint of the arc BC of the circumcircle of ABC not contaning A.
mid-triangle
Let T' = A'B'C' and T" = A"B"C" be two triangles.The mid-triangle of T' and T" is the triangle whose vertices are the midpoints of A'A", B'B" and C'C".
midheight circle
Circumcircle of . ( See Index of triangles referenced in ETC )
Miquel point
See .
Miquel theorem
«In a triangle ABC, if points A', B', C' (all distinct of A, B, C) are chosen on the sidelines BC, CA, AB, respectively, then the circles {{A, B', C'}}, {{B, C', A'}}, {{C, A', B'}} are concurrent ».
The point of concurrence is the Miquel point.
Note: This theorem is also known as pivot theorem.
mittenpunkt
X(9): of the . ( See Index of triangles referenced in ETC ).
mixtilinear circles
Circles internally tangent to the circumcircle and two sidelines of a triangle. Better named mixtilinear incircles.
Circles externally tangent to the circumcircle and two sidelines of a triangle. Called mixtilinear excircles.
mixtilinear excircles
See .
mixtilinear incircles
See .
mixtilinear incircles radical circle
Radical circle of the .
Monge line
Given three circles, their pairwise are collinear on the Monge line.
Morley circles
Circumcircles of . ( See Index of triangles referenced in ETC ). The centers of these circles are named Morley points.
Morley points
See .
Morley-Taylor points
X(357) & X(358).
Moses circle
The Moses circle is defined as the circle with center at the X(39) that is tangent to the . The touchpoint of both circles is X(115).
N
Nagel line
X(1)X(2): Line joining the and the of a triangle.
Nagel point
X(8): of ABC and the . ( See Index of triangles referenced in ETC ).
Napoleon circles
The inner/outer Napoleon circles are the circumcircles of the inner/outer . ( See Index of triangles referenced in ETC ).
Napoleon points
X(17) and X(18)
Neuberg A-, B-, C- circles
See .
Neuberg circles
The 1st A-Neuberg circle is the locus of the vertex A' of a triangle on a given base BC and with a given Brocard angle ω. The same procedure can be repeated for the other two sides of a triangle resulting in a total of three A-, B-, C- 1st Neuberg circles.
Similarly, three reflected Neuberg circles can be obtained from the main circles by reflection in their respective sides of the triangle. These are the A-, B-, C- 2nd Neuberg circles.
The centers of the 1st A-, B-, C-Neuberg circles are the vertices of the . The centers of the 2nd A-, B-, C-Neueberg circles are the vertices of the .
The circumcircles of and are the 1st Neuberg circle and 2nd Neuberg circle. ( See Index of triangles referenced in ETC )
nine-point center
X(5): Center of the .
nine-point circle
Circle through the feet of the altitudes in a triangle. It is also named Feuerbach circle and Euler circle.
Note: This circle passes also through the midpoints of the sides of the triangle and through the midpoints of HA, HB, HC, where H is the orthocenter X(4) of the triangle.The center of this circle is X(5).
O
orthiac triangles
The orthiac triangles of ABC have as vertices the foot of an altitude and the projections of this foot on the other two sides.
Source: TTW - The Triangle Web.
orthic axis
X(230)X(231): of ABC and the . ( See Index of triangles referenced in ETC ).
orthic circle
See .
orthic inconic
Inconic centerd at the X(6).
orthoanticevian triangle
In a triangle ABC with circumcircle Ω, let A'B'C' be the of point P and A", B", C" the respective inverses of A', B', C' in Ω. Points B', C', B", C" lie on a circle ωa with center Oa; Ob and Oc are defined similarly. This circle and Ω are orthogonal.The triangle OaObOc is the orthoanticevian triangle of P.
Source: Preamble before X(8735).
orthoassociate points
Two points are orthoassociate if they are mutually inverse with respect to the .
orthocenter
X(4): Point of intersection of the three of a triangle.
orthocentroidal circle
Circumcircle of the . ( See Index of triangles referenced in ETC )
orthocircumconic
Circumconic of a triangle with in the X(4).
orthocorrespondant
The trilinear pole of the .
orthogonal circles
Two intersecting, non-concentric circles are orthogonal circles when they intersect in two points and their tangents in each of these points are perpendicular.
orthojoin
The of the isogonal conjugate of a point.
orthologic center
See .
orthologic triangles
Let T' = A'B'C', T" = A"B"C" be two triangles. Through A', B', C' draw perpendicular lines (a'), (b'), (c') to B"C", C"A" and A"B", respectively. Through A", B", C" draw perpendicular lines (a"), (b"), (c") to B'C', C'A' and A'B', respectively. It has been proved that if lines (a'), (b'), (c') concur in a point P' then lines (a"), (b"), (c") concur in a point P", and reciprocally. In this case, T' and T" are said to be orthologic triangles and P', P" are called the orthologic centers T' to T" and T" to T', respectively.
orthopoint
A family {ℓ} of parallel lines meet the line at infinity in the same point P. Then all the parallel lines {ℓ'} which are perpendicular to {ℓ} meet the line at in infinity in the same point Q. The point Q is the orthopoint of P.
orthopolar
See .
Source: TTW - The Triangle Web.
orthopole
Let ABC be a triangle and ℓ a line on its plane. Perpendicular lines from A, B, C to ℓ are drawn, meeting ℓ in A', B', C', respectively. Then the perpendicular lines from these points to BC, CA, AB are concurrent in the orthopole of ℓ.
orthoptic circle
The orthoptic circle of a conic 𝒞 is the locus of points from which the tangent lines to 𝒞 are perpendicular.
orthotransversal
Let ABC be a triangle and P a point on its plane. Then the perpendicular lines through P to AP, BP, CP cut BC, CA, AB, respectively, in three collinear points on the orthotransversal or orthopolar of P.
P
parallelian
Any line parallel to a side of a given triangle.
Source: Wolfram's Parallelian.
parallelogic center
See .
parallelogic triangles
Let T' = A'B'C', T" = A"B"C" be two triangles. Through A', B', C' draw parallel lines (a'), (b'), (c') to B"C", C"A" and A"B", respectively. Through A", B", C" draw parallel lines (a"), (b"), (c") to B'C', C'A' and A'B', respectively. It has been proved that if lines (a'), (b'), (c') concur in a point P' then lines (a"), (b"), (c") concur in a point P", and reciprocally. In this case, T' and T" are said to be parallelogic triangles and P', P" are called the parallelogic centers T' to T" and T" to T', respectively.
paralogic triangle
Let ABC be a triangle and 𝓁 a line intersecting BC, CA, AB in A', B', C', respectively. The perpendicular lines to BC, CA, AB through A', B', C', respectively, bound a triangle A"B"C". This last triangle is the paralogic triangle of ABC with respect to 𝓁.
Parry circle
The circle passing through the X(15) and X(16) and the X(2) of a triangle.
Parry point
X(111)
Parry reflection center
X(399): See Forum Geometricorum Volume 8 (2008) 43-48
Pascal line
See .
Pascal theorem
« If a hexagon is inscribed in a (maybe degenerate) conic then the points of intersection of the lines joining the opposite vertices are collinear on the Pascal line, and reciprocally. »
pedal circle
Circumcircle of the of a point.
pedal line
- Let ABC be a triangle and P a point. Any line joining two orthogonal projections of P on the sidelines of ABC is a pedal line.
- Sometimes used as . (for example, in Wikipedia's Pedal triangle). Simson line is preferred for this meaning.
pedal triangle
Let ABC be a triangle, P a point and and A', B', C' the orthogonal projections of P in BC, CA, AB, respectively. Triangle A'B'C' is the pedal triangle of P.
pedal-cevian point
If the of a point P in a triangle ABC is a , then the point P is called a pedal-cevian point.
Source: Wolfram's Pedal-Cevian Point.
perspective center
See .
perspective triangles
Given two triangle A'B'C' and A"B"C", these triangles are said perspective triangles if the lines A'A", B'B", C'C" are concurrent. This point of concurrence is called, center of perspective, homology center or perspector of both triangles, this last being the most used term.
If two perspective triangles are homothetic the perspector is named homothetic center.
When triangles A'B'C' and A"B"C" are perspective, the points A* = B'C'∩B"C" , B* = C'A'∩C"A" and C* = A'B'∩A"B" are collinear on the axis of perspective or perspectrix.
perspector (of a conic)
See .
perspector (of two triangles)
See .
perspectrix
See .
pivot theorem
See .
point-polar triangle
Let ABC be a triangle and A', B', C' three points, none on the sidelines of ABC. The point-polar triangle of A', B', C' is the triangle bounded by the of these points with respect to ABC.
polar circle
Given a triangle ABC, the polar circle of ABC is the circle for which its is ABC itself. This circle is real only when ABC is obtuse. Its center is the X(4).
polar triangle
Let ABC be a triangle and 𝒞 a conic. The polars of A, B, C with respect to 𝒞 bound the polar triangle of ABC with respect to 𝒞.
Poncelot point
Let ABC be a triangle and P a point. The of PBC, PCA, PAB concur at the Poncelot point.
poristic triangles
Two triangles are poristic triangles if they have the same and the same .
Source: TTW - The Triangle Web.
power circles
Circles centered at the midpoints of the sides of a triangle and passing through the opposite vertices.
power point
Triangle centers with triangle center functions of the form f(a) =an.
Prasolov point
X(68): Perspector of ABC and the reflection in the X(5) of the .
R
radical axis
Given two non-concentric circles Ω1 and Ω2, the locus of the points with equal with respect to both circles is a line called the radical axis of Ω1 and Ω2.
Note: It has been proved that, given three non-concentric circles, their pairwise radical axes concur in a point, named the radical center of the circles.
radical center
See .
radical circle
The radical circle of three circles Ω1, Ω2, Ω3 is the to all them. The center of the radical circle of Ω1, Ω2, Ω3 is their .
ratio of homothety
See .
ratio of similarity or similitude
See .
reflection circle
Circumcircle of the . ( See Index of triangles referenced in ETC )
reflection point (of a line)
Given a triangle T and a line 𝓁 the reflection point of 𝓁 is the perspector of T and the triangle bounded by the reflections of 𝓁 in the sidelines of T.
reflection triangle (of a point)
Let ABC be a triangle and P a point. Let A', B', C' be the reflections of P in BC, CA, AB, respectively. A'B'C' is the reflection triangle of P.
Note: Do not confuse with reflection triangle, without any other other specification. This last term refers to the triangle formed by the reflections of A, B, C in BC, CA, AB, respectively. ( See Index of triangles referenced in ETC ).
retrocenter
X(69): of the of X(4).
Rigby points
X(1371): of the and its .
X(1372): of the and its .
( See Index of triangles referenced in ETC ).
S
Schiffler point
X(21): Let ABC be a triangle with I. The Schiffler point is the intersection of the of triangles BCI, CAI and ABI.
Schoute point
X(187): Inverse of the X(6) in the circumcircle.
Schröder point
X(1155): Inverse of X(55) in the circumcircle
Schwatt lines
The Schwatt lines of a triangle are the lines joining the midpoints of the sides and the midpoints of the . They concur at the X(6).
Source: TTW - The Triangle Web.
self-polar triangle
A triangle that is equal to its with respect to a given conic is said to be self-polar or auto-polar with respect to that conic. Any triangle is self-polar with respect to the and .
Source: Wolfram's Self-Polar Triangle.
Sherman line
The side-segments of a triangle are tangent to the incircle and they have their extremities on the circumcircle and their midpoints on the . There exists a fourth segment satisfying these properties. The line containing this segment is the Sherman line of ABC.
Note: The is the line X(3259)X(3326).
Source: Forum Geometricorum Volume 12 (2012) 219-225.
side-triangle
Let T' = A'B'C' and T" = A"B"C" be two triangles. The side-triangle of T' and T" is the triangle T* = A*B*C*, with A* = B'C'∩B"C", B* = C'A'∩C"A" and C* = A'B'∩A"B".
similar figures
In geometry, two geometric figures are said to be similar figures if one can be seen as dilation or contraction of the other.
In the particular case of similar figures having all corresponding lines parallel or confunded, they are said to be homothetic figures. These lines can be those joining coresponding points, or tangents or normal lines through coresponding points. Therefore, the term similar is most frequently reserved for similar figures not being homothetic. In general, homothetic figures are always similar, but similar figures are not always homothetic.
If two similar figures have the same rotational sense they are directly similar (or homothetic) figures, otherwise, they are inverse similar (or homothetic) figures.
If T' and T" are two homothetic figures, there exists at least one point O and a number λ such that, for any pair of corrresponding points P', P" of T' and T", respectively, OP" = λ*OP'. The point O is the homothetic center and the number λ is the ratio or factor of homothety.
If a line passes through the center of homothety of T and T' and cuts T' in P1', P2', ..., then it cuts T" in P1", P2",...., and every pair of corresponding points on T' and T" are named homologus points.
If T' and T" are direct similar figures, there exist a point O, a number λ and an angle θ, such that if T' is rotated by an angle θ around O and then an homothety of center O and factor λ is applied to this rotation, then T" is found. (λ is called ratio of similarity or similitude, θ is the angle of similarity or similitude and O is the center of direct similitude).
If T' and T" are inversely similar, there exist a number λ, a line 𝓁 and a point O on 𝓁, such that if T' is reflected in 𝓁 and then an homothety of center O and factor λ is applied to this reflection, then T" is found. (λ is called ratio of similarity or similitude, 𝓁 is the axis of similarity or similitude and O is the center of inverse similitude).
Any two circles are always homothetic. If they are not-concentric and not-intersecting, they have two centers of similtude, whose names are shortened to exsimilcenterr and insimilcenter, and they are the intersection of the two common external tangents and the intersection of the two common internal tangents, respectively.
Algebraically, and in both cases, each center of similitude is the fixed point in the affine transformation which maps T' into T". (See also ).
similar triangles
See .
Simson line
If P is a point on the circumcircle of a triangle ABC then their orthogonal projections on the sidelines of the triangle are collinear on the Simson line of P.
sine-triple-angle circle
Inscribe two triangles A'B'C' and DeltaA"B"C" in a reference triangle ABC, such that:
∡AB'C' = ∡AC"B" = ∡A,
∡BC'A' = ∡BA"C" = ∡B and
∡CA'B' = ∡CB"A" = ∡C
Then triangles A'B'C' and A"B"C" are inscribed in the sine-triple-angle circle.
sinorthiac triangles
Let ABC be a triangle, Ha the feet of the altitude from A, and Hab, Hac the orthogonal projections of Ha in AC and AB, respectively. The triangle AHabHac is the A-sinorthiac triangle of ABC
Source: TTW - The Triangle Web.
Soddy centers
X(175) and X(176): See .
Soddy circles
In a triangle ABC, the A-, B-, C-Soddy circles are circles centered at A, B, C and pairwise externally tangents.
The inner Soddy circle is the circle externally tangent to each of the . It has center=X(176) and radius=S/(2*(4*R+r+2*s)).
The outer Soddy circle is the circle internally tangent to each of the . It has center=X(175) and radius=S/(2*(4*R+r-2*s)).
X(175) and X(176) are named the outer- and inner- Soddy centers, respectively.
Soddy line
Line X(1)X(7): Line joining the X(1) and the X(7) of a triangle.
Sondat line
If two triangles are and at the same time, (i.e., if they are ), then their and are colliner on the Sondat line.
Spieker center
X(10): Center of the .
Spieker circle
Incircle of the . ( See Index of triangles referenced in ETC ).
Spieker radical circle
of the excircles.
SS-cevian triangle
The SS-cevian triangle of a point X is the of the of X and the point X.
Stammler circle
The Stammler circle is the of the . ( See Index of triangles referenced in ETC ).
Stammler circles
The Stammler circles are the three circles (apart from the circumcircle), that intercept the sidelines of a reference triangle ABC in chords of lengths equal to the corresponding side lengths a, b, and c.
Stammler hyperbola
The Stammler hyperbola of a triangle ABC is the of its . ( See Index of triangles referenced in ETC ).
Steiner circles
The (1st) Steiner circle of a triangle is the circle with center at the X(5) and radius = 3*R/2, wher R is the radius of the circumcircle of the triangle.
The 2nd Steiner circle is the circumcircle of the . ( See Index of triangles referenced in ETC )
Steiner circumellipse
Circumconic centered at the X(2).
Steiner inellipse
Inellipse centered at the X(2).
Steiner line
Let P be a point on the circumcircle of a triangle ABC. The reflections of P on the sidelines of ABC are collinear on the Steiner line.
Steiner point
X(99)
Steiner-Wallace hyperbola
Hyperbola through the X(2), the X(1) and the three excenters of a triangle. Also called simply the Wallace hyperbola.
Source: TTW - The Triangle Web.
Notes:
This hyperbola is the anticomplement of the .
This hyperbola is the locus of points P such that P lies on the line P'P", where P' and P" are the and conjugates of P, respectively.
Stevanovic circle
Circle with center X(650) and squared-radius a*b*c*Σ(a*(a^4-b^4-c^4+a^2*b*c))/(4*Π((b-c))^2))
Stevanovic triangle
Let ABC be a triangle and P a point. Let A', B', C' the same relative centers Q of triangles BCP, CAP and ABP, respectively.
Let A" = (BC)∩(PA') and define B", C" cyclically.
A"B"C" is the Stevanovic triangle of (P, Q)
symgonal point
The symgonal point Q of a point P is equivalently defined as follows :
Source: Bernard Gibert's Cubics in the triangle plane
- if A',B',C' are the reflections of A,B,C in a point P, the circles BCA', CAB', ABC' have a common point Q which is called the symgonal of P. (Jean-Pierre Ehrmann)
- the circumcircles of AB'C', BC'A', CA'B' have a common point Q' which is also on the circumcircle and the circumconic with center P. Q is the reflection of Q' in P. (Paul Yiu)
- Q is the antigonal of the anticomplement of P (or P is the complement of the antigonal of Q).
symmedial circle
The symmedial circle is the of the . ( See Index of triangles referenced in ETC ).
symmedian (line)
In a triangle ABC, the A-symmedian (line) is the eflection of a A- in the internal angle bisector of angle A of ABC.
symmedian point
X(6): Point of concurrence of the .
symmetric triangle
Let ABC be a triangle and P a point. Let A', B', C' be the reflections of P in A, B, C, respectively. A'B'C' is the symmetric triangle of ABC in P.
T
T-cevian triangle
Let t be a central triangle and X a point. The T-cevian triangle of X is the of T and the of the of X, i.e., it is 𝔹(T, cocevian(X-1)).
tangential circle
The tangential circle of a triangle is the of its .
tangential mid-arc circle
The tangential mid-arc circle is the of the . ( See Index of triangles referenced in ETC ).
tangential triangle
In a given a triangle, the tangential triangle is the triangle bounded by the tangent lines to its circumcircle at the vertices of the triangle.
Tarry point
X(98)
Taylor center
X(389): Center of the .
Taylor circle
Let Ha, Hb, Hc be the feet of the of a triangle ABC, on BC, CA, AB, respectively. Let Hab, Hac be the orthogonal projections of Ha in AC and AB, respectively, and build Hbc, Hba, Hca, Hcb cyclically. These six are concyclic and lie on the Taylor circle.
Thebault problem 3
Given any triangle ABC, and any point M on BC, construct the incircle and circumcircle of the triangle. Then construct two additional circles, each tangent to AM, BC, and to the circumcircle. Then their centers and the center of the incircle are collinear.
Sources:
Wikipedia Thébault theorem
Cut-the-Knot Thébault's Problem III
Thomsen hexagon
.
Source: TTW - The Triangle Web.
triangle center
A triangle center (sometimes simply called a center) is a point whose homogeneous coordinates (i.e., barycentrics, trilinear o polar coordinates) are defined in terms of the side lengths and angles of a triangle and for which a triangle center function can be defined.
The triangle center functions of triangles centers therefore satisfy:
1) homogeneity: ƒ(t*a, t*b, t*c) = t^n*ƒ(a, b, c),
2) bisymmetry in b and c: |ƒ(a, c, b)| = |ƒ(a, b, c)|
3) cyclicity in a, b, c, which means that u : v : w = ƒ(a, b, c) : ƒ(b, c, a) : ƒ(c, a, b)
Source: Wolfram's Triangle Center.
Note: When conditions 1 and 3 are satisfied but not 2, the points P1 = ƒ(a, b, c) : ƒ(b, c, a) : ƒ(c, a, b) and P2 = ƒ(a, c, b) : ƒ(b, a, c) : ƒ(c, b, a) are named a bicentric pair.
triangle vertex matrix
Given a triangle, its triangle vertex matrix is a 3x3 matrix whose rows are the homogeneous coordinates of its vertices.
triangulation
Given a triangle ABC and a point P, triangles BCP, CAP and ABP are the triangulation of P.
trilinear coordinates
Let ABC be a triangle and P a point. Let U be the distance of P to line BC, considering U positive if P and A are in the same half-plane with respect to line BC, or negative otherwise. Define V, W similarly, by taking lines CA and AB, respectively. The ordered set {U, V, W} are the actual trilinear coordinates of P. P is univocally determined with these coordinates.
Proportional values to U, V, W can be instead for univocally determining P, i.e., u = k*U, v = k*V, w = k*W, where k≠0 is a constant. In this case u : v : w are the trilinear coordinates of P.
trilinear polar
The of a triangle ABC and the of P is the trilinear polar of P. It is also named the tripolar of 𝓁.
trilinear pole
If 𝓁 is the of a point P, then P is the trilinear pole of 𝓁. It is also named the tripole of 𝓁.
triplicate ratio circle
See .
tripolar
See .
tripolar barycenter
See .
tripolar centroid
With a given triangle, let P be a point and 𝓁 its . Let P' be the point at infinity of 𝓁 and let 𝓁' be the of P'. The point Q = 𝓁 ∩ 𝓁' is the tripolar centroid of P.
Q is also named the tripolar barycenter of P.
tripole
See .
trisected perimeter point
X(369): There exist points A', B', C' on segments BC, CA, AB of atriangle ABC such that B'A + AC' = C'B + BA' = A'C + CB' and lines AA', BB', CC' concur. The point of concurrence is the trisected perimeter point.
Tucker circles
Let ABC be a triangle and A1 a point on BC. Starting from A1 and alternating parallel and lines, follow the next constructions:
1) Through A1 draw a parallel line to AB cutting AC at B1.
2) Through B1 draw an antiparallel line to BC cutting BA at C1.
3) Through C1 draw a parallel line to CA cutting CB at A2.
4) Through A2 draw an antiparallel line to AB cutting AC at B2.
5) Through B2 draw a parallel line to BC cutting BA at C2.
6) Through C2 draw an antiparallel line to CA cutting CB at A3.
Then
1) A3 = A1. This means that no newer points will be found after the step 5 by continuing the sequence. In other words, a closed Tucker hexagon A1B1C1A2B2C2 is obtained always with the last construction.
2) The six points A1, B1, C1, A2, B2, C2 are concyclic and lie on a circle which is said to be a Tucker circle.
Notes:
Same conclusions are achieved if the construction is started with "" instead of "parallel line".
Several named circles are special cases of Tucker circles: .
Tucker hexagon
See .
U
unary cofactor triangle
Let T = A1A2A3 be a central triangle, with Ai = ui : vi : wi, i=1,2,3 (exact trilinear coordinates). Let MT be the of T. The unary cofactor triangle of T, denoted 𝕌(T), is the triangle whose is the cofactor matrix of MT.
V
van Aubel line
Line X(4)X(6): Line joining the and the of a triangle.
van Lamoen circle
The three of a triangle divide it in six triangles. The circumcenters of these triangles lie on the van Lamoen circle.
Vecten circles
The inner/outer Vecten circles are the of the inner/outer . ( See Index of triangles referenced in ETC )
Vecten points
X(485) and X(486): See .
Vecten squares
Given a triangle ABC, build a square in each side, all outwards (or all inwards) ABC. These are the outer- (or inner-) Vecten squares.
If Oa, Ob, Oc are the centers of the outer- (or inner-) Vecten squares, then the lines AOa, BOb, COc concur in the outer- (or inner-) Vecten points, X(485) (or X(486)).
vertex-triangle
Let T' = A'B'C' and T" = A"B"C" be two triangles. The vertex-triangle of T' and T" is the triangle T* = A*B*C*, with A* = B'B"∩C'C", B* = C'C"∩A'A" and C* = A'A"∩B'B".
W
Wabash point
X(364): Equal areas point.
Wallace hyperbola
See .
Weill point
X(354)
Wernau points
X(1337) and X(1338):
Let Fe be the of ABC. Then ABC and Fe are with X(1337) and X(13).
Let Fi be the of ABC. Then ABC and Fi are with X(1338) and X(14).
( See Index of triangles referenced in ETC ).
Source: TTW - The Triangle Web.
Y
Yff central circle
The Yff central circle is the of the . ( See Index of triangles referenced in ETC ).
Yff circles
The Yff circles are the two triplets of congruent circles in which each circle is tangent to two sides of a reference triangle. In each case, the triplets intersect pairwise in a single point.
Yff contact circle
The Yff contact circle is the of the . ( See Index of triangles referenced in ETC ).
Yff hyperbola
Hyperbola having a focus at the X(3) and vertices at the X(2) and the X(4).
Yff parabola
Inconic with perspector X(190).
Yiu circle
The Yiu circle is the of the . ( See Index of triangles referenced in ETC ).
Yiu circles
The A-Yiu-circle of a triangle ABC is the circle passing through vertex A and the reflections of vertices B and C with respect to the opposite sides. B- and C- Yiu circles are defined cyclically.
