## A:

## ABC

The reference triangle

A-vertex coordinates:

Trilinears1 : 0 : 0

## ABC-X

_{3}reflectionsThis triangle is obtained by reflecting A, B, C about the circumcenter X

_{3}. Reference: TCCT 6.12, p. 159.A-vertex coordinates:

Trilinears-1/a : b/SC : c/SB

## AaBbCc and (AaBbCc)*

See

2nd and 4th Przybyłowski-Bollin triangle.

## AiBiCi and (AiBiCi)*

See

1st and 3rd Przybyłowski-Bollin triangle.

## Andromeda

Let A' be the center of the inverse-in-incircle of the A-excircle, and define B' and C' cyclically. The triangle with vertices A,' B, C' is the

Andromeda triangle. (Reference: X5573)A-vertex coordinates:

Trilinears(a^2+3*(b-c)^2)/(3*a^2+(b-c)^2) : 1 : 1

## anticomplementary

The triangle A'B'C' whose medial triangle is ABC. It is also the anticevian-triangle of the centroid of ABC.

A-vertex coordinates:

Trilinears-1/a : 1/b : 1/c

## anti-triangles

The

anti-T triangleis the triangle whose T-triangle is the reference triangle ABC.

## anti-Aquila

The triangle A'B'C' whose Aquila triangle is ABC.

A-vertex coordinates:

Trilinears(2*a+b+c)/a : 1 : 1

## anti-Ara

The triangle A'B'C' whose Ara triangle is ABC.

A-vertex coordinates:

Trilinears(2*SA+SB+SC)/(a*SA) : b/SB : c/SC

## anti-Ascella

The triangle A'B'C' whose Ascella triangle is ABC.

A-vertex coordinates:

Trilinears-a : b*(SA+2*SB)/SB : c*(SA+2*SC)/SC

## 1st anti-Brocard

The triangle A'B'C' whose 1st Brocard triangle is ABC.

A-vertex coordinates:

Trilinears(a^4-b^2*c^2)/a : (c^4-a^2*b^2)/b : (b^4-a^2*c^2)/c

## 4th anti-Brocard

The triangle A'B'C' whose 4th Brocard triangle is ABC.

A-vertex coordinates:

Trilinearsa*((a^2+b^2+c^2)^2-9*b^2*c^2)/(-5*a^2+b^2+c^2) : b*(a^2+b^2-2*c^2) : c*(a^2-2*b^2+c^2)

## 5th anti-Brocard

The triangle A'B'C' whose 5th Brocard triangle is ABC.

A-vertex coordinates:

Trilinears(a^2+c^2)*(a^2+b^2)/a : b^3 : c^3

## 6th anti-Brocard

The triangle A'B'C' whose 6th Brocard triangle is ABC.

A-vertex coordinates:

Trilinears

a*(a^4-b^2*c^2) : (b^6-(a^2+c^2)*b^4-(a^2-c^2)*b^2*c^2+a^2*c^4)/b : (c^6-(a^2+b^2)*c^4-(a^2-b^2)*b^2*c^2+a^2*b^4)/c

## 1st anti-circumperp

The triangle A'B'C' whose 1st circumperp triangle is ABC.

A-vertex coordinates:

Trilinears-a : (b^2-c^2)/b : (c^2-b^2)/c

## (1st) anti-Conway

The triangle A'B'C' whose (1st) Conway triangle is ABC.

A-vertex coordinates:

Trilinearsa^3*(-a^2+b^2+c^2)/((b^2+c^2)*a^2-(b^2-c^2)^2) : b : c

## 2nd anti-Conway

The triangle A'B'C' whose 2nd Conway triangle is ABC.

A-vertex coordinates:

Trilinears((b^2+c^2)*a^2-(b^2-c^2)^2)/a/(-a^2+b^2+c^2) : b : c

## anti-Euler

The triangle A'B'C' whose Euler triangle is ABC.

A-vertex coordinates:

Trilinears(3*a^4-4*(b^2+c^2)*a^2+(b^2-c^2)^2)/(a*(-a^2+b^2+c^2)) : (a^2+b^2-c^2)/b : (a^2-b^2+c^2)/c

## 3rd anti-Euler

The triangle A'B'C' whose 3rd Euler triangle is ABC.

A-vertex coordinates:

Trilinears-((b^2+c^2)*a^2-b^4+b^2*c^2-c^4)*a : (a^4-(b^2-c^2)*(a^2-c^2))*b : (a^4-(c^2-b^2)*(a^2-b^2))*c

## 4th anti-Euler

The triangle A'B'C' whose 4th Euler triangle is ABC.

A-vertex coordinates:

Trilinears

-(-4*R^2*SA-S^2+SA^2-2*SB*SC)*a :

(S^2-2*SA*SC-SB^2+4*(R^2-SC)*SB)*b :

(S^2-2*SA*SB-SC^2+4*(R^2-SB)*SC)*c

## anti-excenters-incenter reflections

The triangle A'B'C' whose excenters-incenter reflections triangle is ABC.

A-vertex coordinates:

Trilinears2*a/(S^2-2*SB*SC) : 1/(b*SB) : 1/(c*SC)

## anti-Hutson intouch

The triangle A'B'C' whose Hutson intouch triangle is ABC.

A-vertex coordinates:

Trilinears-(S^2+2*SA^2)*a : (S^2-2*SA*SB)*b : (S^2-2*SC*SA)*c

## anti-incircle-circles

The triangle A'B'C' whose incircle-circles triangle is ABC.

A-vertex coordinates:

Trilinears-a*(2*S^2+SA^2) : b*(2*S^2-SA*SB) : c*(2*S^2-SA*SC)

## anti-inverse-in-incircle

The triangle A'B'C' whose inverse-in-incircle triangle is ABC.

A-vertex coordinates:

Trilinears-(a^2+b^2+c^2)/a : (a^2+b^2-c^2)/b : (a^2-b^2+c^2)/c

## anti-Mandart-incircle

The triangle A'B'C' whose Mandart-incircle triangle is ABC.

A-vertex coordinates:

Trilinears-a^2+(b+c)*a-2*b*c : (a-b+c)*b : (a+b-c)*c

## anti-McCay

The triangle A'B'C' whose McCay triangle is ABC.

A-vertex coordinates:

Trilinears

-(5*a^4-2*(b^2+c^2)*a^2+(2*b^2-c^2)*(b^2-2*c^2))/a : ((2*a^2+2*b^2-c^2)^2-9*a^2*b^2)/b : ((2*a^2+2*c^2-b^2)^2-9*a^2*c^2)/c

## 6th anti-mixtilinear

The triangle A'B'C' whose 6th mixtilinear triangle is ABC.

A-vertex coordinates:

Trilinears2*a : (a^2-b^2+c^2)/b : (a^2+b^2-c^2)/c

## anti-orthocentroidal

The triangle A'B'C' whose orthocentroidal triangle is ABC.

A-vertex coordinates:

Trilinearsa*((-a^2+b^2+c^2)^2-b^2*c^2) : b*(a^4-(2*b^2-c^2)*a^2+(b^2-c^2)*(b^2+2*c^2)) : c*(a^4+(b^2-2*c^2)*a^2-(b^2-c^2)*(2*b^2+c^2))

## Antlia

Let A' be the center of the inverse-in-A-excircle of the incircle, and define B' and C' cyclically. The triangle A'B'C' is the

Antlia triangleof ABC. (Reference: X5574)A-vertex coordinates:

Trilinears-(a^2+3*(b-c)^2)/(3*a^2+(b-c)^2) : 1 : 1

## Apollonius

The Apollonious circle is the circle internally tangent to all the excircles (Reference). The respective touchpoints are the vertices of the

Apollonius triangle.A-vertex coordinates:

Trilinears-((b+c)*a+b^2+c^2)^2*a/(a+b+c) : (a+c)^2*(a+b-c)*b : (a+b)^2*(a-b+c)*c

## Apus

Let A' be the insimilicenter of the circumcircle and A-excircle, and define B' and C' cyclically. The triangle with vertices A,' B, C' is the

Apus triangle. (Reference: X5584)A-vertex coordinates:

Trilinearsa/(a+b+c) : -b/(a+b-c) : -c/(a-b+c)

## Aquila

Let A' = reflection of the incenter in A, and define B' and C' cyclically. The triangle A'B'C' is the

Aquila triangle. (Reference: X5586)

It is equivalent to the triangle T(1,2) in TCCT, p. 173.A-vertex coordinates:

Trilinears(a+2*b+2*c)/a : -1 : -1

## Ara

Let A'B'C' be the tangential triangle of triangle ABC. Let A" be the center of the A'-excircle of A'B'C', unless this is also the circumcircle of ABC, in which case let A" be the incenter of A'B'C'. Define B", C" cyclically. The triangle A"B"C" is the

Ara triangle. (Reference: X5594)A-vertex coordinates:

Trilinearsa*(a^2+b^2+c^2) : -b*(a^2+b^2-c^2) : -c*(a^2-b^2+c^2)

## Aries

Let A'B'C' be the tangential triangle of an acute triangle ABC. Let A" be the touchpoint of the A-excircle of A'B'C' and the line B'C'; define B" and C" cyclically. The triangle A"B"C" is the

Aries triangle. (Reference: X5596)A-vertex coordinates:

Trilinears-(a^4+(b^2-c^2)^2)/(2*a) : b*(b^2-c^2) : c*(c^2-b^2)

## Artzt

The A-Artzt parabola of a triangle ABC is the parabola tangent at B and C to the sidelines AB and AC, respectively. The triangle A'B'C' bounded by the directrices of the Artzt parabolas is the

Artzt triangle. (Reference: Preamble just before X9742)A-vertex coordinates:

Trilinears-(3*a^4+(b^2-c^2)^2)/(2*a) : ((2*b^2+c^2)*a^2+c^2*(b^2-c^2))/b : ((2*c^2+b^2)*a^2-(b^2-c^2)*b^2)/c

## Ascella

Let A' = incircle-inverse of A, and define B' and C' cyclically. Let O

_{A}be the circle {{B,C,B',C'}}, and define O_{B}and O_{C}cyclically. The three circles are othogonal to the incircle and their centers A",B",C" are the vertices of theAscella triangle. (Reference: Preamble just before X8726)A-vertex coordinates:

Trilinears2*a : (a^2-2*(b+c)*a+c^2-b^2)/b : (a^2-2*(b+c)*a+b^2-c^2)/c

## Atik

Suppose that V is a point outside a circle (U,u). Let (V,v) be the circle with center V that is orthogonal to (U,u), so that v

^{2}= |UV|^{2}- u^{2}= power of V with respect to (U,u). Let U_{V,A}be the circle (V,w) obtained from (U,r) = A-excircle and V = incenter, and let L_{A}be the radical axis of U_{V,A}and the A-excircle. Define L_{B}and L_{C}cyclically. Let A' = L_{B}∩L_{C}, B' = L_{C}∩L_{A}, C' = L_{A}∩L_{B}. The triangle A'B'C' is theAtik triangle. (Reference: Preamble just before X8580))A-vertex coordinates:

Trilinears-((b+c)*a^2-2*(b^2+c^2)*a+(b+c)^3)/a : a^2-2*(b-c)*a+(b+3*c)*(b-c) : a^2-2*(c-b)*a+(c+3*b)*(c-b)

## 1st Auriga

Let U be the inverter (see X5577) of the circumcircle and the incircle. There are two triangles that circumscribe U and are homothetic to triangle ABC, one of which has A-vertex on the same side of line BC as A. This triangle, A'B'C', is the

1st Auriga triangle, and the other, is the 2nd Auriga triangle. (Reference: X5597)A-vertex coordinates:

Trilinearswhere D=sqrt(r*R+4*R^2)(a^4-(b+c)^2*a^2-4*(b+c)*S*D)/a : (b^4-b^2*(a+c)^2+4*b*S*D)/b : (c^4-c^2*(a+b)^2+4*c*S*D)/c

## 2nd Auriga

Let U be the inverter (see X5577) of the circumcircle and the incircle. There are two triangles that circumscribe U and are homothetic to triangle ABC, one of which has A-vertex on the opposite side of line BC as A. This triangle, A"B"C", is the

2nd Auriga triangle, and the other, is the 1st Auriga triangle. (Reference: X5597)A-vertex coordinates:

Trilinearswhere D=sqrt(r*R+4*R^2)(a^4-(b+c)^2*a^2+4*(b+c)*S*D)/a : (b^4-b^2*(a+c)^2-4*b*S*D)/b : (c^4-c^2*(a+b)^2-4*c*S*D)/c

## Ayme

Let R

_{A}be the radical axis of the circumcircle and the A-excircle, and define R_{B}and R_{C}cyclically. Let T_{A}= R_{B}∩R_{C}, and define T_{B}and T_{C}cyclically. (T_{A}is also the radical center of the circumcircle and the B- and C- excircles.) TheAyme triangleis T_{A}T_{B}T_{C}. (Reference: X3610)A-vertex coordinates:

Trilinears(b+c)*(a^2+(b+c)^2)/a : -a^2-b^2+c^2 : -a^2-c^2+b^2

## B: ↑

## BCI

Let I be the incenter of ABC and A', B', C' the incenters of IBC, ICA and IAB, respectively. The triangle A'B'C' is the

BCI triangleof ABC. (Reference)A-vertex coordinates:

Trilinears1 : 1+2*cos(C/2) : 1+2*cos(B/2)

## 1st Brocard

Let Ω' and Ω" be the Brocard points of ABC and A* the intersection of BΩ' and CΩ"; similarly define B* and C*. A*B*C* is the

1st Brocard triangleof ABC. (Reference)A-vertex coordinates:

Trilinearsa*b*c : c^3 : b^3

## 2nd Brocard

Let Ω' and Ω" be the Brocard points of ABC. The circles ω'

_{a}and ω"_{a}through {Ω',B,C} and {Ω",B,C}, respectively, meet again at A*. Define B* and C* similarly. A*B*C* is the2nd Brocard triangleof ABC. (Reference)A-vertex coordinates:

Trilinearsb^2+c^2-a^2 : a*b : a*c

## 3rd Brocard

Let A'B'C' be the 1st Brocard triangle of ABC and A*, B*, C* the isogonal conjugates of A',B',C' with respect to ABC. A*B*C* is the

3rd Brocard triangleof ABC. (Reference)A-vertex coordinates:

Trilinearsb^2*c^2 : a*b^3 : a*c^3

## 4th Brocard

Let A"B"C" be the 2nd Brocard triangle of ABC and A*, B*, C* the isogonal conjugates of A",B",C" with respect to ABC. A*B*C* is the

4th Brocard triangleof ABC. (Reference)A-vertex coordinates:

Trilinearsa*b*c/(b^2+c^2-a^2) : c : b

## 5th Brocard

The

5th Brocard triangleis the vertex triangle of the circumcevian triangles of the 1st and 2nd Brocard points. (Reference: X32)

Note: The vertex triangle of A'B'C' and A"B"C" is the triangle A*B*C*, where A*=B'B"∩C'C" and similarly B* and C* (TCCT, p. 199).A-vertex coordinates:

Trilinears((b^2+c^2)*a^2+(b^2+c^2)^2-b^2*c^2)/a : -b^3 : -c^3

## 6th Brocard

Let Ω' and Ω" be the Brocard points of ABC. The line AΩ' cuts again the 2nd Brocard circle at A' and the line AΩ" cuts again the 2nd Brocard circle at A". Build B',B", C', C" similarly and define A*=B'B"∩C'C", B*=C'C"∩A'A" and C*=A'A"∩B'V". The triangle A*B*C* is the

6th Brocard triangle. (Reference: X384)

Note: The perpendicular bisector of the Brocard points of ABC passes through the circumcenter O of ABC. The circle with center at O and passing through both Brocard points is the 2nd Brocard circle.A-vertex coordinates:

Trilinears((b^2+c^2)*a^2-b^2*c^2)/a : (c^4+c^2*b^2-b^4)/b : (b^4+b^2*c^2-c^4)/c

## C: ↑

## Caelum

Reflections of A,B,C on the incenter. For coordinates, see

5th mixtilinear triangle. (Reference: X5603)

## Carnot

See

Johnson triangle.

## circumcircle-midarc

See

2nd circumperp triangle.

## circummedial

The circumcevian triangle of the centroid of ABC is its

circummedial triangle.A-vertex coordinates:

Trilinears-a*b*c/(b^2+c^2) : c : b

## circumnormal

The

circumnormal triangleis obtained by rotating the circumtangential triangle an angle π/6 around the circumcenter. It is an equilateral triangle. Reference: TCCT, p. 166.A-vertex coordinates:

Trilinearssec(C/3-B/3) : -sec(B/3+2*C/3) : -sec(C/3+2*B/3)

## circumorthic

The circumcevian triangle of the orthocenter of ABC is its

circumorthic triangle.A-vertex coordinates:

Trilinears-a/((b^2+c^2)*a^2-(b^2-c^2)^2) : 1/(b*(a^2-b^2+c^2)) : 1/(c*(a^2+b^2-c^2))

## 1st circumperp

The perpendicular bisector of BC meets the circumcircle in two points A' and A" (A' in the same side of BC as A). Define B', B", C', C" cyclically. The triangle A'B'C' is the

1st circumperp triangle. (TCCT, p. 163).A-vertex coordinates:

Trilinearsa : c-b : b-c

## 2nd circumperp

The perpendicular bisector of BC meets the circumcircle in two points A' and A" (A' in the same side of BC as A). Define B', B", C', C" cyclically. The triangle A"B"C" is the

2nd circumperp triangle. (TCCT, p. 163).A-vertex coordinates:

Trilinears-a : b+c : b+c

## circumsymmedial

The circumcevian triangle of the symmedian point X(6) of ABC is its

circumsymmedial triangle.A-vertex coordinates:

Trilinears-a/2 : b : c

## circumtangential

The

circumtangential trianglehas vertices three points X on the circumcircle of ABC at which the line XX^{-1}is tangent to the circumcircle, where X^{-1}means "the isogonal conjugate of X". It is an equilateral triangle. Reference: TCCT, p. 166 and 168-170.A-vertex coordinates:

Trilinearscsc(C/3-B/3) : csc(B/3+2*C/3) : -csc(C/3+2*B/3)

## contact

See

intouch triangle.

## Conway

Let a, b, c be the sidelengths BC, CA and AB, respectively, of ABC. Let A

_{b}be the point on CA such that AA_{b}= -(a/b)*AC and A_{c}the point on BA such that AA_{c}= -(a/c)*AB. Define B_{c}, B_{a}, C_{a}, C_{b}similarly. TheConway triangleis the triangle bounded by the lines A_{b}A_{c}, B_{c}B_{a}and C_{a}C_{b}. (Reference: X7411)

Note: See this construction at Wolfram's Conway circle.A-vertex coordinates:

Trilinears-(a+b+c)/(b+c) : (a+b-c)/b : (a-b+c)/c

## 2nd Conway

Following the construction of the

Conway triangle, let A_{2}= B_{a}B_{c}∩C_{a}C_{b}, and define B_{2}and C_{2}cyclically. The triangle A_{2}B_{2}C_{2}is the2nd Conway triangleof ABC. (Reference: Preamble just before X9776)A-vertex coordinates:

Trilinears-(a+b+c)/a : (a+b-c)/b : (a-b+c)/c

## 3rd Conway

Following the construction of the

Conway triangle, let A_{3}be the intersection of the tangents to the Conway circle at B_{a}and C_{a}, and define B_{3}, C_{3}cyclically. The triangle A_{3}B_{3}C_{3}is the3rd Conway triangle. (Reference: X10434)

Note: The six points A_{b}, A_{c}, B_{c}, B_{a}, C_{a}, C_{b}lie on a circle named the Conway circle.A-vertex coordinates:

Trilinears

-a*(a+b+c)^2 : ((b+2*c)*a^3+2*a^2*b^2+(b-c)*(2*c^2+b*c+b^2)*a+2*(b^2-c^2)*b*c)/b : ((2*b+c)*a^3+2*a^2*c^2-(b-c)*(c^2+b*c+2*b^2)*a-2*(b^2-c^2)*b*c)/c

## 4th Conway

Following the construction of the

Conway triangle, let A_{4}be the intersection of the tangents to the Conway circle at A_{b}and A_{c}, and define B_{4}, C_{4}cyclically. The triangle A_{4}B_{4}C_{4}is the4th Conway triangle. (Reference: X10434)

Note: The six points A_{b}, A_{c}, B_{c}, B_{a}, C_{a}, C_{b}lie on a circle named the Conway circle.A-vertex coordinates:

Trilinears(a^3-(b+c)*a^2-2*(b^2+b*c+c^2)*a-2*b*c*(b+c))/((a+b+c)*a^2) : 1 : 1

## 5th Conway

Following the construction of the

Conway triangle, let A_{5}be the intersection of the tangents to the Conway circle at B_{c}and C_{b}, and define B_{5}, C_{5}cyclically. Triangle A_{5}B_{5}C_{5}is named the5th Conway triangle. (Reference: X10434)

Note: The six points A_{b}, A_{c}, B_{c}, B_{a}, C_{a}, C_{b}lie on a circle named the Conway circle.A-vertex coordinates:

Trilinears-((b+c)*a^2+(b^2+c^2)*a+2*b*c*(b+c))/((a+b+c)*(b+c)*a) : 1 : 1

## inner-Conway

Let a, b, c be the sidelengths BC, CA and AB, respectively, of ABC. Let A

_{b}be the point on CA such that AA_{b}= (a/b)*AC and A_{c}the point on BA such that AA_{c}= (a/c)*AB. Define B_{c}, B_{a}, C_{a}, C_{b}similarly. Theinner-Conway triangleis the triangle bounded by the lines A_{b}A_{c}, B_{c}B_{a}and C_{a}C_{b}. (Reference: Preamble just before X11677)

Notes: See this construction at Wolfram's Conway circle.

Points A_{b}, A_{c}, B_{c}, B_{a}, C_{a}and C_{b}do not lie on a conic.A-vertex coordinates:

Trilinears-1 : (b-c)/b : (c-b)/c

## D: ↑

## D-triangle

See

4th Brocard triangle.

## de Villiers

See

BCI triangle.

## E: ↑

## 1st Ehrmann

In Hyacinthos #6098, December 2, 2002, Jean-Pierre Ehrmann defines a circle as follows. Let P be a point in the plane of ABC and not on the lines BC, CA, AB. Let A

_{B}the the point of intersection of the circle {{P,B,C}} and the line AB. Define A_{C}symmetrically, and define B_{C}, B_{A}, C_{A}, C_{B}cyclically. These six points of intersection are on a circle if and only if P = X(6). The centers A_{1}, B_{1}, C_{1}of the circles {{X(6),B,C}}, {{X(6),C,A}}, {{X(6),A,B,}}, respectively, are the vertices of the1st Ehrmann triangle. (Reference: Preamble just before X8537)A-vertex coordinates:

Trilinears-(a^4-b^4+4*b^2*c^2-c^4)*a : (a^2*(a^2+5*c^2)-b^4+3*b^2*c^2-2*c^4)*b : (a^2*(a^2+5*b^2)-2*b^4+3*b^2*c^2-c^4)*c

## 2nd Ehrmann

Following the construction of the

1st Ehrmann triangle, let A_{2}= B_{C}B_{A}∩C_{A}C_{B}, and define B_{2}and C_{2}cyclically. The triangle A_{2}B_{2}C_{2}is the2nd Ehrmann triangle. (Reference: Preamble just before X8537)A-vertex coordinates:

Trilinearsa*(a^2-2*b^2-2*c^2)/(2*a^2-b^2-c^2) : b : c

## Euler

Let H be the orthocenter of ABC and A*, B*, C* the midpoints of AH, BH and CH, respectively. A*B*C* is the

Euler triangleof ABC. (Reference)

A*, B*, C* are also the reflections of the vertices of the medial triangle in X(5).A-vertex coordinates:

Trilinears2((b^2+c^2)*a^2-(b^2-c^2)^2)/(a*(b^2+c^2-a^2)) : (a^2+b^2-c^2)/b : (a^2-b^2+c^2)/c

## 2nd Euler

The reflections of the vertices of the orthic triangle in X(5) are the vertices of the

2nd Euler triangle. (Reference: X3758)A-vertex coordinates:

Trilinears-2*((b^2+c^2)*a^2-(b^2-c^2)^2)*a : (a^2-b^2+c^2)*(a^4-2*a^2*c^2+(b^2-c^2)^2)/b : (a^2+b^2-c^2)*(a^4-2*a^2*b^2+(b^2-c^2)^2)/c

## 3rd Euler

The perpendicular to BC through X(5) cuts the nine point circle in two points A', A", with A' farther from A than A". Build B', B", C', C" similarly. A'B'C' is the

3rd Euler triangle. (Reference: X3758)A-vertex coordinates:

Trilinears-(b-c)^2/a : (a^2+c*(b-c))/b : (a^2+b*(c-b))/c

## 4th Euler

The perpendicular to BC through X(5) cuts the nine-points-circle in two points A', A", with A" closer to A than A'. Build B', B", C', C" similarly. A"B"C" is the

4th Euler triangle. (Reference: X3758)A-vertex coordinates:

Trilinears-(b+c)^2/a : (a^2-c*(b+c))/b : (a^2-b*(b+c))/c

## 5th Euler

The nine-points-circle cuts again the median AA' in A", where A' is the midpoint of BC. If B" and C" are built similarly then A"B"C" is the

5th Euler triangle. (Reference: X3758)A-vertex coordinates:

Trilinears-2*(b^2+c^2)/(a*(a^2-b^2-c^2)) : 1/b : 1/c

## excenters-incenter midpoints

The A-vertex is the midpoint of the A-excenter and the incenter.

Note: The original name of this triangle isT(-2,1). Reference: TCCT, p. 173.A-vertex coordinates:

Trilinears(-2*a+b+c)/a : 1 : 1

## excenters-incenter reflections

The A-vertex is the reflection of the A-excenter in the incenter.

Note: The original name of this triangle isT(-1,3). Reference: TCCT, p. 173.A-vertex coordinates:

Trilinears(-a+3*b+3*c)/(-3*a+b+c) : 1 : 1

## excentral

The triangle whose vertices are the excenters of ABC.

A-vertex coordinates:

Trilinears-1 : 1 : 1

## extangents

The triangle A'B'C' formed by the points of pairwise intersection of the three exterior tangents to the excircles.

A-vertex coordinates:

Trilinears-a*(a+b+c) : (a+c)*(a+b-c) : (a+b)*(a-b+c)

## extouch

Let A' be the point where the A-excircle touches the side BC and define B', C' similarly. A'B'C' is the

extouch triangleof ABC.A-vertex coordinates:

Trilinears0 : 1/(b*(a+b-c)) : 1/(c*(a-b+c))

## 2nd extouch

Let A

_{A},A_{B}, A_{C}be the touchpoints of the A-excircle and the lines BC, CA, AB, respectively, and define B_{B}, B_{C}, B_{A}and C_{C}, C_{A}, C_{B}cyclically.

Let A_{2}= B_{C}B_{A}∩C_{A}C_{B}, and define B_{2}and C_{2}cyclically. The2nd extouch triangleis A_{2}B_{2}C_{2}. (Reference: X5927)A-vertex coordinates:

Trilinears-2*(b+c) : (a^2+b^2-c^2)/b : (a^2-b^2+c^2)/c

## 3rd extouch

Let A

_{A},A_{B}, A_{C}be the touchpoints of the A-excircle and the lines BC, CA, AB, respectively, and define B_{B}, B_{C}, B_{A}and C_{C}, C_{A}, C_{B}cyclically.

Let A_{3}= C_{A}A_{C}∩A_{B}B_{A}, and define B_{3}and C_{3}cyclically. The3rd extouch triangleis A_{3}B_{3}C_{3}. (Reference: X5927)A-vertex coordinates:

Trilinears-2*(b+c)*(a+b-c)*(a-b+c)/((-a+b+c)*(a+b+c)) : (a^2+b^2-c^2)/b : (a^2-b^2+c^2)/c

## 4th extouch

Let A

_{A},A_{B}, A_{C}be the touchpoints of the A-excircle and the lines BC, CA, AB, respectively, and define B_{B}, B_{C}, B_{A}and C_{C}, C_{A}, C_{B}cyclically.

Let A_{4}= B_{C}C_{A}∩B_{A}C_{B}, and define B_{4}and C_{4}cyclically. The4th extouch triangleis A_{4}B_{4}C_{4}. (Reference: X5927)A-vertex coordinates:

Trilinears-2*(b+c)*(a+b+c)/(-a+b+c) : (a^2-b^2+c^2)/b : (a^2+b^2-c^2)/c

## 5th extouch

Let A

_{A},A_{B}, A_{C}be the touchpoints of the A-excircle and the lines BC, CA, AB, respectively, and define B_{B}, B_{C}, B_{A}and C_{C}, C_{A}, C_{B}cyclically.

Let D_{1}= B_{B}B_{C}∩C_{C}C_{A}, and define D_{2}and D_{3}cyclically. Let E_{1}= B_{B}B_{A}∩C_{C}C_{B}, and define E_{2}and E_{3}cyclically. Define A_{5}= D_{2}E_{2}∩D_{3}E_{3}, and define B_{5}and C_{5}cyclically. The5th extouch triangleis A_{5}B_{5}C_{5}. (Reference: X5927)A-vertex coordinates:

Trilinears-2*(b+c)/(-a+b+c) : (b^2+c^2+a^2+2*c*a)/(b*(a-b+c)) : (b^2+2*a*b+a^2+c^2)/(c*(a+b-c))

## F: ↑

## Feuerbach

The

Feuerbach triangleis the triangle formed by the three points of tangency of the nine-point circle with the excircles. (Reference)A-vertex coordinates:

Trilinears-(b-c)^2*(a+b+c)/a : (a+c)^2*(a+b-c)/b : (a+b)^2*(a-b+c)/c

## Fuhrmann

The

Fuhrmann trianglehas vertices the reflections of the vertices of 2nd circumperp triangle on the sidelines of ABC.A-vertex coordinates:

Trilinearsa : -(a^2-c*(b+c))/b : -(a^2-b*(b+c))/c

## G: ↑

## inner-Garcia

Let T

_{A}T_{B}T_{C}be the extouch triangle of a triangle ABC, and let L_{A}be the line perpendicular to line BC at T_{A}. Of the two points on L_{A}at distance r=inradius-of-ABC from T_{A}, let A' be the one farther from A and let A" be the closer. Define B', C' and B", C" cyclically. A'B'C' is the outer Garcia triangle and A"B"C" is theinner Garcia triangle. (Reference: X5587)A-vertex coordinates:

Trilinearsa : (c*a-b^2+c^2)/b : (a*b+b^2-c^2)/c

## outer-Garcia

Let T

_{A}T_{B}T_{C}be the extouch triangle of a triangle ABC, and let L_{A}be the line perpendicular to line BC at T_{A}. Of the two points on L_{A}at distance r=inradius-of-ABC from T_{A}, let A' be the one farther from A and let A" be the closer. Define B', C' and B", C" cyclically. A'B'C' is theouter Garcia triangleand A"B"C" is the inner Garcia triangle. (Reference: X5587)A-vertex coordinates:

Trilinears-1 : (a+c)/b : (a+b)/c

## Gossard

The Euler line of ABC cuts BC, CA, AB at A',B',C', respectively. The Euler lines of AB'C', BC'A' and CA'B' enclose the

Gossard triangle. (Reference: X402)A-vertex coordinates:

Trilinears

SA^2*(SB-SC)^2*(S^2-3*SB*SC)/(a*S^2) :

(S^2-3*SA*SC)*(4*R^2*(-6*SB+SW)+5*S^2-SW^2+6*SB^2-4*SA*SC)/b :

(S^2-3*SA*SB)*(4*R^2*(-6*SC+SW)+5*S^2-SW^2+6*SC^2-4*SA*SB)/c

## inner-Grebe

Erect squares BB

_{a}C_{a}C, CC_{b}A_{b}A, AA_{c}B_{c}B on the sides BC, CA, AB of triangle ABC (inwards). Then, the triangle A'B'C' enclosed by the lines B_{a}C_{a}, C_{b}A_{b}, A_{c}B_{c}is theinner-Grebe triangleof ABC.A-vertex coordinates:

Trilinears-(b^2+c^2-S)/a : b : c

## outer-Grebe

Erect squares BB

_{a}C_{a}C, CC_{b}A_{b}A, AA_{c}B_{c}B on the sides BC, CA, AB of triangle ABC (outwards). Then, the triangle A'B'C' enclosed by the lines B_{a}C_{a}, C_{b}A_{b}, A_{c}B_{c}is theouter-Grebe triangleof ABC. (Reference: Hyacinthos #6445)A-vertex coordinates:

Trilinears-(b^2+c^2+S)/a : b : c

## H: ↑

## half-altitude

See

midheight triangle.

## 1st Hatzipolakis

Let A'B'C' be the intouch triangle of ABC. Let C

_{A}be the point other than C' in which the perpendicular to BC from C' meets the incircle, let B_{A}be the point other than B' in which the perpendicular to BC from B' meets the incircle, and let A_{0}be the point of intersection of lines BC_{A}and CB_{A}. Define B_{0}and C_{0}cyclically. A_{0}B_{0}C_{0}is the1st Hatzipolakis triangle. (Reference: X1118)A-vertex coordinates:

Trilinears(-a+b+c)*a : (a^2+b^2-c^2)^2/(b*(a-b+c)) : (a^2-b^2+c^2)^2/(c*(a+b-c))

## 2nd Hatzipolakis

Let A

_{0}B_{0}C_{0}be the 1st Hatzipolakis triangle of ABC. Let A_{1}be the orthogonal projection of A_{0}onto line BC, and define B_{1}and C_{1}cyclically. The triangle A_{1}B_{1}C_{1}is the2nd Hatzipolakis triangle. (Reference: X1119)A-vertex coordinates:

Trilinears0 : 1/(b*(a^2-b^2+c^2)*(a-b+c)^2) : 1/(c*(a^2+b^2-c^2)*(a+b-c)^2)

## hexyl

Given a triangle ABC with excenters J

_{a}, J_{b}, J_{c}, define A* as the point in which the perpendicular to AB through the excenter J_{b}meets the perpendicular to AC through the excenter J_{c}, and similarly define B* and C*. Then A*B*C* is thehexyl triangleof ABC. (Reference)A-vertex coordinates:

Trilinearsa^3+(-b-c)*a^2-(b+c)^2*a+(b^2-c^2)*(b-c) : a^3+(c-b)*a^2-(b-c)^2*a+(b^2-c^2)*(b+c) : a^3+(b-c)*a^2-(b-c)^2*a-(b^2-c^2)*(b+c)

## Honsberger

Let A'B'C' be the intouch triangle of ABC. Let L

_{A}be the line through X(7) parallel to B'C', and let A_{B}= AB∩L_{A}and A_{C}= AC∩L_{A}. Define B_{C}and C_{A}cyclically, and define B_{A}and C_{B}cyclically. (The six points A_{B}, B_{C}, C_{A}, A_{C}, B_{A}, C_{B}lie on the Adams circle). Let A* = A_{B}C_{B}∩A_{C}B_{C}, and define B* and C* cyclically. The triangle A*B*C* is theHonsberger triangle. (Reference: X7670)A-vertex coordinates:

Trilinears1/((b+c)*a-(b-c)^2) : -1/(b*(a-b+c)) : -1/(c*(a+b-c))

## Hung-Feuerbach

Let F

_{a}F_{b}F_{c}be the Feuerbach triangle of ABC and (K_{a}) the circle, other than the nine-point-circle, passing through F_{b}and F_{c}and tangent to the A-excircle (I_{a}). Let A' be the touchpoint of (K_{a}) and (I_{a}) and name as t_{a}the tangent to both circles in A'. Build t_{b}and t_{c}cyclically. The triangle bounded by t_{a}, t_{b}and t_{c}is named here theHung-Feuerbach triangleof ABC. (Reference: X5973)A-vertex coordinates:

Trilinears

-(b+c)^2*(a+b-c)*(a-b+c)*(a^2+(b+c)^2)*(a^2+c*a+b*(b-c))*(a^2+b*a+3*(c-b))/a :

(a+c)*(a^5+c*a^4-(b^2-c^2)^2*a+c*(b^2-c^2)*(3*b^2+c^2))*((a+c)*b+a^2+c^2)*((-a+b)*c+a^2+b^2)/b :

(a+b)*(a^5+b*a^4-(c^2-b^2)^2*a+b*(c^2-b^2)*(3*c^2+b^2))*((a+b)*c+a^2+b^2)*((-a+c)*b+a^2+c^2)/c

## inner-Hutson

The internal bisector of angle A meets the A-excircle in two points. Let P

_{A}be the point closer to line BC and let Q_{A}be the other point. Define P_{B}and P_{C}cyclically, and define Q_{B}and Q_{C}cyclically. Let L_{A}be the line tangent to the A-excircle at P_{A}, and define L_{B}and L_{C}cyclically. Theinner-Hutson triangleis the triangle bounded by the lines L_{A}, L_{B}and L_{C}. (Reference: X363)A-vertex coordinates:

Trilinearswhere x = (-a+b+c)/2, y = (a-b+c)/2, z = (a+b-c)/2 and δ

-(2*δ^{2}*sin(A/2)-y^{2}*sin(B/2)-z^{2}*sin(C/2)) :

(2*(c*z-δ^{2})*b*sin(A/2)-(a-c)*y^{2}*sin(B/2)-b*z^{2}*sin(C/2))/b :

(2*(b*y-δ^{2})*c*sin(A/2)-(a-b)*z^{2}*sin(C/2)-c*y^{2}*sin(B/2))/c

^{2}= (4*R*r+r^{2})/2

## outer-Hutson

Continuing the construction of the inner-Hutson triangle, let M

_{A}be the line tangent to the A-excircle at Q_{A}, and define M_{B}and M_{C}cyclically. Theouter-Hutson triangleis the triangle bounded by the lines M_{A}, M_{B}and M_{C}. (Reference: X363)A-vertex coordinates:

Trilinearswhere x = (-a+b+c)/2, y = (a-b+c)/2, z = (a+b-c)/2 and δ

-(2*δ^{2}*sin(-A/2)-y^{2}*sin(B/2)-z^{2}*sin(C/2)) :

(2*(c*z-δ^{2})*b*sin(-A/2)-(a-c)*y^{2}*sin(B/2)-b*z^{2}*sin(C/2))/b :

(2*(b*y-δ^{2})*c*sin(-A/2)-(a-b)*z^{2}*sin(C/2)-c*y^{2}*sin(B/2))/c

^{2}= (4*R*r+r^{2})/2

## Hutson extouch

Let A' be the antipode of the A-extouch point in the A-excircle, and define B' and C' cyclically. A'B'C' is the

Hutson-extouch triangle. (Reference: X5731)A-vertex coordinates:

Trilinears-4*a/(a+b+c) : (a+b-c)/b : (a-b+c)/c

## Hutson intouch

Let A' be the antipode of the A-intouch point in the incircle, and define B' and C' cyclically. A'B'C' is the

Hutson-intouch triangle. (Reference: X5731)A-vertex coordinates:

Trilinears4*a/(-a+b+c) : (a-b+c)/b : (a+b-c)/c

## 1st Hyacinth

Let A'B'C' be the orthic triangle of a triangle ABC, and

O = circumcenter of ABC

Ab = reflection of A' in BO, and define Bc and Ca cyclically

Ac = reflection of A' in CO, and define Ba and Cb cyclically

(Na) = nine-point circle of A'AbAc, and define (Nb) and (Nc) cyclically

Na = nine-point center of A'AbAc, and define Nb and Nc cyclically

The triangle NaNbNc is the1st Hyacinth triangle. (Reference: X10111)A-vertex coordinates:

Trilinears

-a^3*SA*(SA^2-3*S^2)/(2*S^2) :

((12*R^2-2*SW)*S^2-(2*R^2*(SA+SC)-SC^2+S^2-(SA+SC)^2)*SB)/b :

((12*R^2-2*SW)*S^2-(2*R^2*(SA+SB)-SB^2+S^2-(SA+SB)^2)*SC)/c

## 2nd Hyacinth

Let A'B'C' be the orthic triangle of ABC. Define A* as the orthogonal projection of A on B'C', and B* and C* similarly. A*B*C* es the

2nd Hyacinth triangleof ABC.

(Reference: Hyacinthos #24020)A-vertex coordinates:

Trilinears-((b^2+c^2)*a^4-2*(b^2-c^2)^2*a^2+(b^2+c^2)*(b^2-c^2)^2)/(a*(a^2-b^2-c^2)) : (a^2-b^2+c^2)*b : (a^2+b^2-c^2)*c

## I: ↑

## incentral

Cevian triangle of the incenter.

A-vertex coordinates:

Trilinears0 : 1 : 1

## incircle-circles

Let a', b', c' be the incircle-inverses of the sidelines BC, CA and AB, respectively of ABC and A', B', C' their centers. A'B'C' is the

incircle-circles triangleof ABC. (Reference: X11034)A-vertex coordinates:

Trilinears2*a : (a^2+4*a*b+b^2-c^2)/b : (a^2+4*a*c+c^2-b^2)/c

## intangents

The triangle A'B'C' formed by the points of pairwise intersection of the three interior tangents to the excircles.

A-vertex coordinates:

Trilinears-a*(-a+b+c) : (a-c)*(a-b+c) : (a-b)*(a+b-c)

## intouch

The

intouch triangleof ABC, also called thecontact triangle, has as vertices the touchpoints of the incircle and the sidelines of ABC.A-vertex coordinates:

Trilinears0 : 1/(b*(a-b+c)) : 1/(c*(a+b-c))

## inverse-in-incircle

The

inverse-in-incircle triangleof ABC has as vertices the inverses of A,B,C in the incircle of ABC. (Reference: X11018)A-vertex coordinates:

Trilinears((b+c)*a-(b-c)^2)/(a*(-a+b+c)) : 1 : 1

## J: ↑

## Johnson

The vertices of the

Johnson triangleare the centers of the Johnson circles J_{a}, J_{b}and J_{c}. (Reference)

Explanation: Johnson's theorem states that if three equal circles mutually intersect one another in a single point, then the circle J passing through their other three pairwise points of intersection is congruent to the original three circles. If J_{a}is the circle {{B,C,X(4)}} and J_{b}and J_{c}are defined cyclically, then each one has radius=circumradius and J is the circumcircircle of ABC.

Note: The Johnson triangle is also known asCarnot triangle.A-vertex coordinates:

Trilinears-a*(a^2-b^2-c^2) : (a^4+(-b^2-2*c^2)*a^2-c^2*(b^2-c^2))/b : (a^4+(-2*b^2-c^2)*a^2+(b^2-c^2)*b^2)/c

## inner-Johnson

Of the two tangents to the J

_{b}and J_{c}Johnson circles of ABC (seeJohnson triangle), let L_{A}be the one on the opposite side of BC from A, and let M_{A}be the other. Define L_{B}, M_{B}, L_{C}, M_{C}cyclically. Theinner-Johnson triangleis the triangle enclosed by the lines L_{A}, L_{B}and L_{C}. (Reference: X10522)A-vertex coordinates:

Trilinears-a^2+(b+c)*a-2*b*c : (a-b+c)*(a-c)^2/b : (a+b-c)*(a-b)^2/c

## outer-Johnson

Of the two tangents to the J

_{b}and J_{c}Johnson circles of ABC (seeJohnson triangle), let L_{A}be the one on the opposite side of BC from A, and let M_{A}be the other. Define L_{B}, M_{B}, L_{C}, M_{C}cyclically. Theouter-Johnson triangleis the triangle enclosed by the lines M_{A}, M_{B}and M_{C}. (Reference: X10522)A-vertex coordinates:

Trilinears-a^2-(b+c)*a-2*b*c : (a+c)^2*(a+b-c)/b : (a+b)^2*(a-b+c)/c

## 1st Johnson-Yff

The Yff circles are the two triplets of congruent circles in which each circle is tangent to two sides of a reference triangle (see Yff circles at Wolfram's). The circles Y'

_{a}, Y'_{b}and Y'_{c}of the first triplet concur at X(55) and the circles Y"_{a}, Y"_{b}and Y"_{c}of the second triplet concur at X(56). The vertices of the1st Johnson-Yff triangleare the other points of intersection of Y'_{a}, Y'_{b}and Y'_{c}. (Reference: X10523).

Note: By Johnson theorem, if three congruent circles concur then the circle through their other points of intersection is congruent to the original circles. This is the reason for the name of the triangle.A-vertex coordinates:

Trilinearsa/(-a+b+c) : (a+c)^2/(b*(a-b+c)) : (a+b)^2/(c*(a+b-c))

## 2nd Johnson-Yff

The Yff circles are the two triplets of congruent circles in which each circle is tangent to two sides of a reference triangle (see Yff circles at Wolfram's). The circles Y'

_{a}, Y'_{b}and Y'_{c}of the first triplet concur at X(55) and the circles Y"_{a}, Y"_{b}and Y"_{c}of the second triplet concur at X(56). The vertices of the2nd Johnson-Yff triangleare the other points of intersection of Y"_{a}, Y"_{b}and Y"_{c}. (Reference: X10523).

Note: By Johnson theorem, if three congruent circles concur then the circle through their other points of intersection is congruent to the original circles. This is the reason for the name of the triangle.A-vertex coordinates:

Trilinears-a*(-b-c+a) : (a-b+c)*(a-c)^2/b : (a+b-c)*(a-b)^2/c

## K: ↑

## 1st Kenmotu diagonals

Kenmotu squares of ABC are three equal squares such that each square touches two sides of ABC and all three squares touch at a single common point (Reference). There are two triplets of such squares: in the first of them the squares have X(371) as common vertex, in the second triplet the common vertex is X(372). The diagonals not through the common point of the squares in first triplet surround the

1st Kenmotu diagonal triangle.A-vertex coordinates:

Trilinearsa*(b^2+c^2-a^2-2*S)/(b^2+c^2-a^2+2*S) : b : c

## 2nd Kenmotu diagonals

Kenmotu squares of ABC are three equal squares such that each square touches two sides of ABC and all three squares touch at a single common point (Reference). There are two triplets of such squares: in the first of them the squares have X(371) as common vertex, in the second triplet the common vertex is X(372). The diagonals not through the common point of the squares in second triplet surround the

2nd Kenmotu diagonal triangle.A-vertex coordinates:

Trilinearsa*(b^2+c^2-a^2+2*S)/(b^2+c^2-a^2-2*S) : b : c

## Kosnita

The vertices of the

Kosnita triangleare the circumcenters of the triangles BOC, COA, AOB, where O is the circumcenter of ABC. (Reference: X1658)A-vertex coordinates:

Trilinears(a^4+(-2*b^2-2*c^2)*a^2+b^4+c^4)*a : -b*(a^4+(-2*b^2-c^2)*a^2+(b^2-c^2)*b^2) : -(a^4+(-b^2-2*c^2)*a^2-c^2*(b^2-c^2))*c

## L: ↑

## Lemoine

The

Lemoine triangleis the cevian triangle of X(598).A-vertex coordinates:

Trilinears0 : a*c/(2*a^2+2*c^2-b^2) : a*b/(2*a^2+2*b^2-c^2)

## Lucas triangles

Build outwards ABC the square BCC

_{A}B_{A}(see figure). Lines AB_{A}and AC_{A}cut BC at A'_{B}and A'_{C}, respectively, and the perpendiculars through these points cut AB and AC at A"_{C}and A"_{B}, respectively. The quadrilateral A'_{B}A'_{C}A"_{B}A"_{C}is a square named theA-Lucas square(and also theA-inner-inscribed-square). The B- and C- Lucas squares, and the points B"_{C}, B"_{A}, C"_{A}, C"_{B}can be built similarly.

The circles {{A,A"_{B},A"_{C}}}, {{B,B"_{C},B"_{A}}} and {{C,C"_{A},C"_{B}}} are theA-, B-, C- Lucas circlesand result to be pairwise tangents and simultaneously tangent internally to the circumcircle of ABC. The circle externally tangent to the Lucas circles is named theLucas inner circle.

The other circle tangent to the circumcircle and to the B- and C- Lucas circles is named theA-Lucas secondary central circle, and the B- and C-Lucas secondary central circles are defined in the same way.

Denote:

A_{0}= the center of the A-Lucas circle, and similarly B_{0}, C_{0};

A_{T}= the touchpoint of the B- and C- Lucas circles, and similarly B_{T}, C_{T};

A_{p}= the antipode of A in the A-Lucas circle, and similarly B_{p}, C_{p};

A_{i}= the touchpoint of the A- Lucas circle and the Lucas inner circle, and similarly B_{i}, C_{i};

A_{2}= the center of the A-Lucas secundary central circle, and similarly B_{2}, C_{2};

A_{2O}= the touchpoints of the A-Lucas secundary central circle and the circumcircle, and similarly B_{2O}, C_{2O};

A_{2B}, A_{2C}= the touchpoints of the A-Lucas secundary central circle and the B- and C- Lucas circles, respectively, and similarly B_{2C}, B_{2A}, C_{2A}, C_{2B}. These six point lie on theLucas secondary tangents circle.

If the construction is made with squares drawn inwards ABC then all the above terms can be reused just by replacing the nameLucaswithLucas(-1); examples:A-Lucas(-1) squareandA-Lucas(-1) circleinstead ofA-Lucas squareandA-Lucas circle. Moreover, if the construction is generalized and BCC_{A}B_{A}is a rectangle whose sides are in the proportion length:width=t:1 (i.e, BB_{A}= CC_{A}= t*BC), then all the terms described can be reused just by replacingLucasbyLucas(t), assuming t>0 for rectangles built outwards ABC and t<0 otherwise. In this case we must be awared that Lucas(t) circles are pairwise tangent if, and only if, t=1 or t=-1.

## Lucas antipodal

Vertices: A

_{p}, B_{p}and C_{p}. See Lucas triangles. (Reference: X6457)A-vertex coordinates:

Trilinearscos(B)*cos(C)-sin(A) : -cos(B) : -cos(C)

## Lucas(-1) antipodal

Vertices: A

_{p}, B_{p}and C_{p}. See Lucas triangles. (Reference: X6457)A-vertex coordinates:

Trilinearscos(B)*cos(C)+sin(A) : -cos(B) : -cos(C)

## Lucas Brocard

The

Lucas Brocard triangleis bounded by the radical axis of the Brocard circle and the Lucas circles. See Lucas triangles. (Reference: X6421)A-vertex coordinates:

Trilinears-a*(a^2-2*S) : b*(a^2+c^2-S) : c*(a^2+b^2-S)

## Lucas(-1) Brocard

The

Lucas(-1) Brocard triangleis bounded by the radical axis of the Brocard circle and the Lucas(-1) circles. See Lucas triangles. (Reference: X6421)A-vertex coordinates:

Trilinears-a*(a^2+2*S) : b*(a^2+c^2+S) : c*(a^2+b^2+S)

## Lucas central

Vertices: A

_{0}, B_{0}and C_{0}. See Lucas triangles.A-vertex coordinates:

Trilinearscos(A)+2*sin(A) : cos(B) : cos(C)

## Lucas(-1) central

Vertices: A

_{0}, B_{0}and C_{0}. See Lucas triangles.A-vertex coordinates:

Trilinearscos(A)-2*sin(A) : cos(B) : cos(C)

## Lucas homothetic

The

Lucas homothetic triangleis enclosed by the lines A"_{B}A"_{C}, B"_{C}B"_{A}and C"_{B}C"_{B}. See Lucas triangles. (Reference: X493)A-vertex coordinates:

Trilinears-(b^2+c^2-a^2)^2/(4*a) : b*(c^2+S) : c*(b^2+S)

## Lucas(-1) homothetic

The

Lucas(-1) homothetic triangleis enclosed by the lines A"_{B}A"_{C}, B"_{C}B"_{A}and C"_{B}C"_{B}. See Lucas triangles. (Reference: X493)A-vertex coordinates:

Trilinears-(b^2+c^2-a^2)^2/(4*a) : b*(c^2-S) : c*(b^2-S)

## Lucas inner

Vertices: A

_{i}, B_{i}and C_{i}. See Lucas triangles.A-vertex coordinates:

Trilinears2*cos(A)+3/2*sin(A) : 2*cos(B)+sin(B) : 2*cos(C)+sin(C)

## Lucas(-1) inner

Vertices: A

_{i}, B_{i}and C_{i}. See Lucas triangles.A-vertex coordinates:

Trilinears2*cos(A)-3/2*sin(A) : 2*cos(B)-sin(B) : 2*cos(C)-sin(C)

## Lucas inner tangential

This triangle is bounded by the tangents to the Lucas inner circle at A

_{i}, B_{i}and C_{i}. See Lucas triangles. (Reference: Preamble just before X6395)A-vertex coordinates:

Trilinears4*cos(A)+sin(A) : 4*cos(B)+3*sin(B) : 4*cos(C)+3*sin(C)

## Lucas(-1) inner tangential

This triangle is bounded by the tangents to the Lucas(-1) inner circle at A

_{i}, B_{i}and C_{i}. See Lucas triangles. (Reference: Preamble just before X6395)A-vertex coordinates:

Trilinears4*cos(A)-sin(A) : 4*cos(B)-3*sin(B) : 4*cos(C)-3*sin(C)

## Lucas reflection

Let L

_{a}be the reflection of the line B_{0}C_{0}in BC and define L_{b}, L_{c}cyclically (see Lucas triangles). TheLucas reflection triangleis the triangle limited by L_{a}, L_{b}, L_{c}. (Reference: X6401)A-vertex coordinates:

Trilinears

-cos(B)*cos(C)*sin(A)^2+(cos(B)+(1+sin(2*B))*cos(A-C)+sin(B))*(cos(C)+(1+sin(2*C))*cos(A-B)+sin(C)) :

cos(B)*(cos(C)*sin(A)*sin(B)-(cos(C)+(1+sin(2*C))*cos(A-B)+sin(C))*sin(C)) :

cos(C)*(cos(B)*sin(A)*sin(C)-(cos(B)+(1+sin(2*B))*cos(A-C)+sin(B))*sin(B))

## Lucas(-1) reflection

Let L

_{a}be the reflection of the line B_{0}C_{0}in BC and define L_{b}, L_{c}cyclically (see Lucas triangles). TheLucas(-1) reflection triangleis the triangle limited by L_{a}, L_{b}, L_{c}. (Reference: X6401)A-vertex coordinates:

Trilinears

-cos(B)*cos(C)*sin(A)^2+(cos(B)+(1-sin(2*B))*cos(A-C)-sin(B))*(cos(C)+(1-sin(2*C))*cos(A-B)-sin(C)) :

cos(B)*(cos(C)*sin(A)*sin(B)+(cos(C)+(1-sin(2*C))*cos(A-B)-sin(C))*sin(C)) :

cos(C)*(cos(B)*sin(A)*sin(C)+(cos(B)+(1-sin(2*B))*cos(A-C)-sin(B))*sin(B))

## Lucas secondary central

Vertices: A

_{2}, B_{2}and C_{2}. See Lucas triangles. (Reference: X6199)A-vertex coordinates:

Trilinearscos(A)-2*sin(A) : cos(B)+4*sin(B) : cos(C)+4*sin(C)

## Lucas(-1) secondary central

Vertices: A

_{2}, B_{2}and C_{2}. See Lucas triangles. (Reference: X6199)A-vertex coordinates:

Trilinearscos(A)+2*sin(A) : cos(B)-4*sin(B) : cos(C)-4*sin(C)

## Lucas 1st secondary tangents

Triangle enclosed by the lines B

_{2A}C_{2A}, C_{2B}A_{2B}and A_{2C}B_{2C}(see Lucas triangles). (Reference: X6199)A-vertex coordinates:

Trilinearscos(A)-2*sin(A) : cos(B)+3*sin(B) : cos(C)+3*sin(C)

## Lucas(-1) 1st secondary tangents

Triangle enclosed by the lines B

_{2A}C_{2A}, C_{2B}A_{2B}and A_{2C}B_{2C}(see Lucas triangles). (Reference: X6199)A-vertex coordinates:

Trilinearscos(A)+2*sin(A) : cos(B)-3*sin(B) : cos(C)-3*sin(C)

## Lucas 2nd secondary tangents

Triangle enclosed by the lines A

_{2B}A_{2C}, B_{2C}B_{2A}and C_{2A}C_{2B}(see Lucas triangles). (Reference: X6199)A-vertex coordinates:

Trilinearscos(A)+6*sin(A) : cos(B)-sin(B) : cos(C)-sin(C)

## Lucas(-1) 2nd secondary tangents

Triangle enclosed by the lines A

_{2B}A_{2C}, B_{2C}B_{2A}and C_{2A}C_{2B}(see Lucas triangles). (Reference: X6199)A-vertex coordinates:

Trilinearscos(A)+6*sin(A) : cos(B)-sin(B) : cos(C)-sin(C)

## Lucas tangents

Vertices: A

_{T}, B_{T}and C_{T}. See Lucas triangles.A-vertex coordinates:

Trilinearscos(A) : cos(B)+sin(B) : cos(C)+sin(C)

## Lucas(-1) tangents

Vertices: A

_{T}, B_{T}and C_{T}. See Lucas triangles.A-vertex coordinates:

Trilinearscos(A) : cos(B)-sin(B) : cos(C)-sin(C)

## M: ↑

## MacBeath

The

MacBeathtriangle is the triangle whose vertices are the contact points of the MacBeath inconic with the reference triangle ABC. (Reference).

Note: The MacBeath inconic of a triangle is the inconic having foci the orthocenter and the circumcenter, giving the center as the nine-point center.A-vertex coordinates:

Trilinears0 : 1/(b^3*(a^2-b^2+c^2)) : 1/(c^3*(a^2+b^2-c^2))

## Mandart-excircles

Let A'B'C' be the intangents triangle of ABC and A* the point where B'C' touchs the A-excircle of ABC. Build B* and C* cyclically. The triangle A*B*C* is the

Mandart-excircles triangleof ABC. (Reference: X10974)A-vertex coordinates:

Trilinears-(b-c)^2*(a+b+c)/a : b*(a+b-c) : c*(a-b+c)

## Mandart-incircle

Let A'B'C' be the intangents triangle of ABC and A* the point where B'C' touchs the incircle of ABC. Build B* and C* cyclically. The triangle A*B*C* is the

Mandart-incircle triangleof ABC.A-vertex coordinates:

Trilinears(b+c-a)*(b-c)^2/a, b*(a-b+c), c*(a+b-c)

## McCay

The

McCay triangleis the triangle whose vertices are the centers of the McCay circles. (Reference: X7606)

Note: Let G be the centroid and A'B'C' the2nd Brocard triangleof ABC. The A-, B- and C- McCay circles of ABC are the circles {{G,B',C'}}, {{G,C',A'}} and {{G,A',B'}}, respectively. (Reference)A-vertex coordinates:

Trilinearsa*(a^2+b^2+c^2) : -(2*a^4+(-3*c^2-2*b^2)*a^2+(b^2-c^2)*(-c^2+2*b^2))/b : -(2*a^4+(-2*c^2-3*b^2)*a^2+(b^2-c^2)*(-2*c^2+b^2))/c

## medial

The

medial triangleis the cevian triangle of the centroid.A-vertex coordinates:

Trilinears0 : 1/b : 1/c

## midarc

The A-angle bisector cuts the incircle at two points A' and A", with A closer from A than A". If B', B", C' C" are defined similarly then A'B'C' is the

midarc triangleof ABC. (Reference)A-vertex coordinates:

Trilinears(cos(B/2)+cos(C/2))^2 : cos(A/2)^2 : cos(A/2)^2

## 2nd midarc

The A-angle bisector cuts the incircle at two points A' and A", with A closer from A than A". If B', B", C' C" are defined similarly then A"B"C" is the

2nd midarc triangleof ABC.A-vertex coordinates:

Trilinears(cos(B/2)-cos(C/2))^2 : cos(A/2)^2 : cos(A/2)^2

## midheight

Let A'B'C' be the orthic triangle of ABC and A*, B*, C* the midpoints of AA', BB', CC', respectively. A*B*C* is the

midheight triangleof ABC. (It is also known as thehalf-altitude triangle)A-vertex coordinates:

Trilinears2*a : (a^2+b^2-c^2)/b : (a^2-b^2+c^2)/c

## mixtilinear

The

mixtilinear triangleis the triangle connecting the centers of themixtilinear incircles. It is alson known as theinner-mixtilinear triangle. (Reference)

Note: A circle that isinternallytangent to two sides of a triangle and to the circumcircle is called amixtilinear incircle. There are three mixtilinear incircles, one corresponding to each angle of the triangle (Reference). Also, there are three circles that areexternallytangent to two sides of a triangle and to the circumcircle and are called amixtilinear excircles.A-vertex coordinates:

Trilinears-(a^3+(b+c)*a^2-(b+c)^2*a-(b^2-c^2)*(b-c))/(4*a*b*c) : 1 : 1

## 2nd mixtilinear

The triangle whose vertices are the centers of the

mixtilinear excirclesis the2nd mixtilinear triangle. It is alson known as theouter-mixtilinear triangle. (See note in mixtilinear triangle). (Reference: Preamble just before X7955)A-vertex coordinates:

Trilinears(a^3+-(b+c)*a^2-(b+c)^2*a+(b^2-c^2)*(b-c))/(4*a*b*c) : 1 : 1

## 3rd mixtilinear

The

3rd mixtilinear trianglehas as vertices the touchpoints of the circumcircle and the mixtilinear incircles. (See note in mixtilinear triangle). (Reference: Preamble just before X7955)A-vertex coordinates:

Trilinears-1 : 2*b/(a-b+c) : 2*c/(a+b-c)

## 4th mixtilinear

The

4th mixtilinear trianglehas as vertices the touchpoints of the circumcircle and the mixtilinear excircles. (See note in mixtilinear triangle). (Reference: Preamble just before X7955)A-vertex coordinates:

Trilinears-1 : 2*b/(a+b-c) : 2*c/(a-b+c)

## 5th mixtilinear

Let I

_{ab}and I_{ac}be the touchpoints of the A-mixtilinear incircle and the sidelines AC and AB, respectively, and define I_{bc}, I_{ba}, I_{ca}, I_{cb}similarly. Denote A_{5}=I_{ab}I_{ba}∩I_{ac}I_{ca}, B_{5}=I_{bc}I_{cb}∩I_{ba}I_{ab}, C_{5}=I_{ca}I_{ac}∩I_{cb}I_{bc}. A_{5}B_{5}C_{5}is the5th mixtilinear triangle. (See note in mixtilinear triangle).

Note: This triangle is also namedCaelum triangle. (Reference: Preamble just before X7955)A-vertex coordinates:

Trilinears-(b+c-a)/(2*a) : 1 : 1

## 6th mixtilinear

Let E

_{ab}and E_{ac}be the touchpoints of the A-mixtilinear excircle and the sidelines AC and AB, respectively, and define E_{bc}, E_{ba}, E_{ca}, E_{cb}similarly. The6th mixtilinear triangleis the triangle bounded by the lines E_{ab}E_{ac}, E_{bb}E_{ba}and E_{ca}E_{cb}. (See note in mixtilinear triangle). (Reference: Preamble just before X7955)A-vertex coordinates:

Trilinearsa^2-2*(b+c)*a+(b-c)^2 : a^2-(b-c)*(2*a-b-3*c) : a^2-(c-b)*(2*a-c-3*b)

## 7th mixtilinear

Let E

_{ab}and E_{ac}be the touchpoints of the A-mixtilinear excircle and the sidelines AC and AB, respectively, and define E_{bc}, E_{ba}, E_{ca}, E_{cb}similarly. Denote A_{7}=E_{ab}E_{ba}∩E_{ac}E_{ca}, B_{7}=E_{bc}E_{cb}∩E_{ba}E_{ab}, C_{7}=E_{ca}E_{ac}∩E_{cb}E_{bc}. A_{7}B_{7}C_{7}is the7th mixtilinear triangle. (See note in mixtilinear triangle). (Reference: X8916)A-vertex coordinates:

Trilinears(a-b+c)*(a+b-c)*(a^2-2*(b+c)*a+(b-c)^2)/(2*(-a+b+c)*a) : a^2-2*(b-c)*a+(b+3*c)*(b-c) : a^2+2*(b-c)*a+(c+3*b)*(c-b)

## inner-mixtilinear

See

mixtilinear triangle.

## outer-mixtilinear

See

2nd mixtilinear triangle.

## Montesdeoca-Hung

Let (Ap) be the Apollonius circle and (K

_{a}), (K_{b}), (K_{c}) the circles defined in theHung-Feuerbach triangle. Let L_{a}be the radical axis of (Ap) and (K_{a}) and define L_{b}and L_{c}similarly. The triangle enclosed by these lines is theMontesdeoca-Hung triangle. (Reference: X6042)

Note: The Apollonious circle is the circle internally tangent to all the excircles (Reference)A-vertex coordinates:

Trilinears

2*a^5*(a+2*b+2*c)+4*a^3*(b^2+3*b*c+c^2)*(a+b+c)+(3*b^4+3*c^4+4*b*c*(3*b^2+5*b*c+3*c^2))*a^2+(b+c)^2*(2*(b+c)*(b^2+c^2)*a+b^4+c^4) :

-(a+c)^2*(a^2+b*a+c*(b+c))^2 :

-(a+b)^2*(a^2+c*a+b*(c+b))^2

## 1st Morley

Let b

_{a}and b_{c}be the two angle trisectors of B, and c_{a}and c_{b}the two angle trisectors of C, with b_{a}and c_{a}closer to BC than the others. Build a_{b}and a_{c}similarly. Define A_{1}= b_{a}∩c_{a}and similarly B_{1}and C_{1}. The triangle A_{1}B_{1}C_{1}is equilateral and is named the1st Morley triangle. (Reference)A-vertex coordinates:

Trilinears1 : 2*cos(C/3) : 2*cos(B/3)

## 2nd Morley

Suppose that ABC is labeled counter-clockwise. With the same notation used in the 1st Morley triangle, rotate b

_{a}around B an angle Pi/3 (counter-clockwise) and let b'_{a}be the resulting line. Next rotate c_{a}around C an angle Pi/3 (clockwise) and denote c'_{a}the resulting line. Define A_{2}= b'_{a}∩c'_{a}and similarly B_{2}and C_{2}. The triangle A_{2}B_{2}C_{2}is equilateral and is named the2nd Morley triangle. (Reference)

Note: Invert the senses of rotation if ABC is labeled clockwise.A-vertex coordinates:

Trilinears-1 : 2*cos(C/3+π/3) : 2*cos(B/3+π/3)

## 3rd Morley

Suppose that ABC is labeled counter-clockwise. With the same notation used in the 1st Morley triangle, rotate b

_{a}around B an angle Pi/3 (clockwise) and let b"_{a}be the resulting line. Next rotate c_{a}around C an angle Pi/3 (counter-clockwise) and denote c"_{a}the resulting line. Define A_{3}= b"_{a}∩c"_{a}and similarly B_{3}and C_{3}. The triangle A_{3}B_{3}C_{3}is equilateral and is named the3rd Morley triangle. (Reference)

Note: Invert the senses of rotation if ABC is labeled clockwise.A-vertex coordinates:

Trilinears-1 : 2*cos(C/3-π/3) : 2*cos(B/3-π/3)

## 1st Morley-adjunct

With the same notation used in 1st Morley triangle, denote A'

_{1}= b_{c}∩c_{b}and similarly B'_{1}and C'_{1}. The triangle A'_{1}B'_{1}C'_{1}is the1st Morley-adjunct triangle. (Reference)

Note: The 1st Morley-adjunct triangle is not equilateral.A-vertex coordinates:

Trilinears2 : sec(C/3) : sec(B/3)

## 2nd Morley-adjunct

Suppose that ABC is labeled counter-clockwise. With the same notation used in the 1st Morley triangle, rotate b

_{c}around B an angle Pi/3 (counter-clockwise) and let b'_{c}be the resulting line. Next rotate c_{b}around C an angle Pi/3 (clockwise) and denote c'_{b}the resulting line. Define A'_{2}= b'_{c}∩c'_{b}and similarly B'_{2}and C'_{2}. The triangle A'_{2}B'_{2}C'_{2}is equilateral is the2nd Morley-adjunct triangle. (Reference)

Note: Invert the senses of rotation if ABC is labeled clockwise. This triangle is not equilateral.A-vertex coordinates:

Trilinears-2 : sec(C/3+π/3) : sec(B/3+π/3)

## 3rd Morley-adjunct

Suppose that ABC is labeled counter-clockwise. With the same notation used in the 1st Morley triangle, rotate b

_{c}around B an angle Pi/3 (clockwise) and let b"_{a}be the resulting line. Next rotate c_{b}around C an angle Pi/3 (counter-clockwise) and denote c"_{a}the resulting line. Define A'_{3}= b"_{c}∩c"_{b}and similarly B'_{3}and C'_{3}. The triangle A'_{3}B'_{3}C'_{3}is the3rd Morley-adjunct triangle. (Reference)

Note: Invert the senses of rotation if ABC is labeled clockwise. This triangle is not equilateral.A-vertex coordinates:

Trilinears-2 : sec(C/3-π/3) : sec(B/3-π/3)

## Moses-Hung

Let H

_{a}H_{b}H_{c}be the orthic triangle of ABC and (J_{a}) the circle, other than the nine-point-circle, passing through H_{b}and H_{c}and tangent to the A-excircle (I_{a}). Let A' be the touchpoint of (J_{a}) and (I_{a}) and define B' and C' cyclically. The triangle A'B'C' is named here theMoses-Hung triangleof ABC. (Reference: X6044)A-vertex coordinates:

Trilinears-(2*a^3+(b+c)*a^2+(b^2-c^2)*(b-c))^2/(a*(a+b+c)) : (a+c)^2*(a+b-c)^3/b : (a+b)^2*(a-b+c)^3/c

## N: ↑

## inner-Napoleon

The

inner Napoleon triangleis the triangle A"B"C" formed by the centers of internally erected equilateral triangles BCA', CAB' and ABC' on the sides of a given triangle ABC. It is an equilateral triangle. (Reference)A-vertex coordinates:

Trilinears-a : (SC-sqrt(3)*S)/b : (SB-sqrt(3)*S)/c

## outer-Napoleon

The

outer Napoleon triangleis the triangle A"B"C" formed by the centers of externally erected equilateral triangles BCA', CAB' and ABC' on the sides of a given triangle ABC. It is an equilateral triangle. (Reference)A-vertex coordinates:

Trilinears-a : (SC+sqrt(3)*S)/b : (SB+sqrt(3)*S)/c

## 1st Neuberg

The triangle A

_{1}B_{1}C_{1}formed by joining the centers of the Neuberg circles is the1st Neuberg triangle. (Reference)

Note: The Neuberg A-circle is the locus of A of a triangle on a given base BC and with a given Brocard angle ω. (Reference)A-vertex coordinates:

Trilinears-a*SW : (SC^2-SA*SB)/b : (SB^2-SA*SC)/c

## 2nd Neuberg

The triangle A

_{2}B_{2}C_{2}formed by joining the centers of the reflections of the Neuberg circles on the opposite sides of ABC is the2nd Neuberg triangle. (Reference)A-vertex coordinates:

Trilinears-a*SW : (SC^2-SA*SB+2*S^2)/b : (SB^2-SA*SC+2*S^2)/c

## O: ↑

## orthic

Cevian and pedal triangle of X(4).

A-vertex coordinates:

Trilinears0 : 1/(b*(a^2-b^2+c^2)) : 1/(c*(a^2+b^2-c^2))

## orthocentroidal

Let A' be the intersection, other than X(4), of the A-altitude and the orthocentroidal circle, and define B' and C' cyclically. The triangle A'B'C'is the

orthocentroidal triangle. (Reference: X5476)

Note: The orthocentroidal circle has diameter X(2)X(4).A-vertex coordinates:

Trilinearsa : (a^2+b^2-c^2)/b : (a^2-b^2+c^2)/c

## 1st orthosymmedial

Let A' be the intersection, other than the orthocenter, of the A-altitude and the circle with diameter X(4)X(6). Define B' and C' similarly. The triangle A'B'C' is the

1st orthosymmedial triangle.A-vertex coordinates:

Trilinears2*a^3/(b^2+c^2) : (a^2+b^2-c^2)/b : (a^2-b^2+c^2)/c

## 2nd orthosymmedial

Let A" be the intersection, other than the symmedian center, of the A-symmedian and the circle with diameter X(4)X(6). Define B" and C" similarly. The triangle A"B"C" is the

2nd orthosymmedial triangle.A-vertex coordinates:

Trilinears-((b^4+c^4)*a^2+(b^2-c^2)^2*(-b^2-c^2))/(a*(b^2+c^2)*(a^2-b^2-c^2)) : b : c

## P: ↑

## 1st Pamfilos-Zhou

The Pamfilos-Zhou A-rectangle R

_{A}= AA_{B}A_{A}A_{C}is the rectangle of maximal area such that A is a vertex of R_{A}, B lies on the line A_{A}A_{C}, and C lies on the line A_{A}A_{B}. The Pamfilos-Zhou B- and C-rectangles, R_{B}and R_{C}, are defined cyclically.

Let A' = B_{C}B_{A}∩C_{A}C_{B}, and define B' and C' cyclically. The1st Pamfilos-Zhoutriangle is A'B'C'. (Reference: X7594)A-vertex coordinates:

Trilinears

-a^4*(a^2+b*c)+(b^2+c^2)*(b+c)^2*a^2+b*c*(b+c)*(4*S*a-(b^2-c^2)*(b-c)) :

(a^2-b^2+c^2)*((a^2+b^2+c^2)*a*b+2*c^2*S) :

(a^2+b^2-c^2)*((a^2+b^2+c^2)*a*c+2*b^2*S)

## 2nd Pamfilos-Zhou

The Pamfilos-Zhou A-rectangle R

_{A}= AA_{B}A_{A}A_{C}is the rectangle of maximal area such that A is a vertex of R_{A}, B lies on the line A_{A}A_{C}, and C lies on the line A_{A}A_{B}. The Pamfilos-Zhou B- and C-rectangles, R_{B}and R_{C}, are defined cyclically.

Let A" = C_{A}A_{C}∩A_{B}B_{A}, and define B" and C" cyclically. The2nd Pamfilos-Zhoutriangle is A"B"C". (Reference: X7594)A-vertex coordinates:

Trilinears

-2*(b+c)*S-(-a+b+c)*((b+c)*a+(b-c)^2) :

(2*(a-c)*a*S+(-a+b+c)*(a^2*c+(b-c)*(b^2+a*b+c^2)))/b :

(2*(a-b)*a*S+(-a+b+c)*(a^2*b+(c-b)*(c^2+a*c+b^2)))/c

## 1st Parry

Let A

_{1}be the intersection, other than X(111), of the Parry circle and the line AX(111), and define B_{1}and C_{1}cyclically. The triangle A_{1}B_{1}C_{1}is the1st Parry triangle. (Reference: Preamble just before X9123)

Note: TheParry circleis the circle passing through the isodynamic points X(15) and X(16) and the centroid X(2) of a triangle.A-vertex coordinates:

Trilinears3*a^4+(-2*b^2-2*c^2)*a^2-b^2*c^2+c^4+b^4 : -b*(a^2+b^2-2*c^2)*a : -c*(a^2-2*b^2+c^2)*a

## 2nd Parry

Let A

_{2}be the intersection, other than X(110), of the Parry circle and line AX(110), and define B_{2}and C_{2}cyclically. The triangle A_{2}B_{2}C_{2}is the2nd Parry triangle. (Reference: Preamble just before X9123)

Note: TheParry circleis the circle passing through the isodynamic points X(15) and X(16) and the centroid X(2) of a triangle.A-vertex coordinates:

Trilinears(a^4+b^2*c^2-b^4-c^4)*(b^2-c^2) : (a^2-b^2)*b*a*(2*a^2-b^2-c^2) : -(a^2-c^2)*c*a*(2*a^2-b^2-c^2)

## 3rd Parry

Let A

_{3}be the intersection, other than X(2), of the Parry circle and the A-median, and define B_{3}and C_{3}cyclically. The triangle A_{3}B_{3}C_{3}, is the3rd Parry triangle. (Reference: Preamble just before X9123)

Note: TheParry circleis the circle passing through the isodynamic points X(15) and X(16) and the centroid X(2) of a triangle.A-vertex coordinates:

Trilinearsa*(a^4+(-3*b^2-3*c^2)*a^2+2*b^4+b^2*c^2+2*c^4) : b*c^2*(2*a^2-b^2-c^2) : b^2*c*(2*a^2-b^2-c^2)

## Pelletier

The

Pelletier triangleis the vertex triangle of ABC and theintangets triangle. Reference: TCCT, p. 162.

Note: Thevertex triangleA*B*C* of triangles T_{1}=A_{1}B_{1}C_{1}and T_{2}=A_{2}B_{2}C_{2}has vertex A*=B_{1}B_{2}∩C_{1}C_{2}, and B*, C* are defined cyclically.A-vertex coordinates:

Trilinears-(b-c)*(-a+b+c), (c-a)*(a-b+c), (a-b)*(a-c+b)

## 1st Przybyłowski-Bollin

Let D

_{1}be the 1st isodynamic center X(15) of ABC and I'_{a}, I'_{b}and I'_{c}the incenters of BCD_{1}, CAD_{1}and ABD_{1}, respectively. The triangle I'_{a}I'_{b}I'_{c}is the1st Przybyłowski-Bollin triangle. (Reference: Preamble just before X11752)

Note: This triangle is denoted asAiBiCiin ETC.A-vertex coordinates:

Trilinearsa*(sqrt(3)*SA+S) : sqrt(3)*SB*b+(b+U)*S : sqrt(3)*SC*c+(c+U)*S

where U = 2*sqrt(SW+sqrt(3)*S)

## 2nd Przybyłowski-Bollin

Let D

_{1}be the 1st isodynamic center X(15) of ABC and J'_{a}the excenter of BCD_{1}opposed to D_{1}. Define J'_{b}and J'_{c}similarly. The triangle J'_{a}J'_{b}J'_{c}is the2nd Przybyłowski-Bollin triangle. (Reference: Preamble just before X11752)

Note: This triangle is denoted asAaBbCbin ETC.A-vertex coordinates:

Trilinearsa*(sqrt(3)*SA+S) : sqrt(3)*b*SB+(b-U)*S : sqrt(3)*c*SC+(c-U)*S

where U = 2*sqrt(SW+sqrt(3)*S)

## 3rd Przybyłowski-Bollin

Let D

_{2}be the 2nd isodynamic center X(16) of ABC and I"_{a}, I"_{b}and I"_{c}the incenters of BCD_{2}, CAD_{2}and ABD_{2}, respectively. The triangle I"_{a}I"_{b}I"_{c}is the3rd Przybyłowski-Bollin triangle. (Reference: Preamble just before X11752)

Note: This triangle is denoted as(AiBiCi)*in ETC.A-vertex coordinates:

Trilinearsa*(sqrt(3)*SA-S) : sqrt(3)*SB*b-(b+V)*S : sqrt(3)*SC*c-(c+V)*S

where V = 2*sqrt(SW-sqrt(3)*S)

## 4th Przybyłowski-Bollin

Let D

_{2}be the 2nd isodynamic center X(16) of ABC and J"_{a}the excenter of BCD_{2}opposed to D_{2}. Define J"_{b}and J"_{c}similarly. The triangle J"_{a}J"_{b}J"_{c}is the4th Przybyłowski-Bollin triangle. (Reference: Preamble just before X11752)

Note: This triangle is denoted as(AaBbCb)*in ETC.A-vertex coordinates:

Trilinearsa*(sqrt(3)*SA-S) : sqrt(3)*SB*b-(b-V)*S : sqrt(3)*SC*c-(c-V)*S

where V = 2*sqrt(SW-sqrt(3)*S)

## Q: ↑

## R: ↑

## reflection

Let A' be the reflection of A in BC and define B', C' similarly. A'B'C' is the

reflection triangleof ABC.A-vertex coordinates:

Trilinears-1 : (a^2+b^2-c^2)/(a*b) : (a^2-b^2+c^2)/(a*c)

## Roussel

If the angular trisectors of a triangle are produced to the circumcircle, then the chords of adjacent trisectors form an equilateral triangle named the

Roussel triangle. (Reference)A-vertex coordinates:

Trilinears

-a^2-2*(c*cos(B/3)+b*cos(C/3))*a+4*b*c*(4*cos(A/3)^2-1)*cos(B/3)*cos(C/3) :

(a+2*b*cos(C/3))*(2*cos(A/3)*c+b)+4*c*cos(A/3)*cos(B/3)*(2*cos(A/3)*b+c) :

(a+2*c*cos(B/3))*(2*cos(A/3)*b+c)+4*b*cos(A/3)*cos(C/3)*(2*cos(A/3)*c+b)

## S: ↑

## 1st Schiffler

It is well known that if P lies on the Neuberg cubic then the Euler lines L

_{a}, L_{b}and L_{c}of PBC, PCA and PAB, respectively, are concurrent. In particular, for P=X(1) and P=X(3065), the point of concurrence is the Schiffler center S=X(21).

Let P=X(1) and B_{c}the point of intersection, other than S, of L_{b}and the circle (C,|CS|) and C_{b}the point of intersection, other than S, of L_{c}and the circle (B,|BS|). Define O'_{a}as the circumcenter of SB_{c}C_{b}and similarly O'_{b}and O'_{c}. The triangle O'_{a}O'_{b}O'_{c}is the1st Schiffler triangle. (Reference: X6595)A-vertex coordinates:

Trilinears1/a : (a-c)/(a^2-(b+2*c)*a+c^2-b^2) : (a-b)/(a^2-(c+2*b)*a+b^2-c^2)

## 2nd Schiffler

It is well known that if P lies on the Neuberg cubic then the Euler lines L

_{a}, L_{b}and L_{c}of PBC, PCA and PAB, respectively, are concurrent. In particular, for P=X(1) and P=X(3065), the point of concurrence is the Schiffler center S=X(21).

Let P=X(3065) and B_{c}the point of intersection, other than S, of L_{b}and the circle (C,|CS|) and C_{b}the point of intersection, other than S, of L_{c}and the circle (B,|BS|). Define O"_{a}as the circumcenter of SB_{c}C_{b}and similarly O"_{b}and O"_{c}. The triangle O"_{a}O"_{b}O"_{c}is the2nd Schiffler triangle. (Reference: X6596)A-vertex coordinates:

Trilinears-1/a : (a-c)/(a^2+(b-2*c)*a+c^2-b^2) : (a-b)/(a^2+(c-2*b)*a+b^2-c^2)

## Schröeter

Let A'B'C' and A"B"C" be the medial and orthic triangles of ABC, respectively. Define A*=B'C'∩B"C" and similarly B* and C*. A*B*C* is the

Schröeter triangleof ABC.A-vertex coordinates:

Trilinears1/a : (a^2-c^2)/(b*(b^2-c^2)) : (a^2-b^2)/(c*(c^2-b^2))

## 1st Sharygin

Let A', B', C' be the feet of the internal angles bisectors. The perpendicular bisectors of AA', BB', CC' bound a triangle A*B*C* called the

1st Sharygin triangle. (Reference)A-vertex coordinates:

Trilinears-a^2+b*c : a*b+c^2 : a*c+b^2

## 2nd Sharygin

Let A", B", C" the feet of the external angles bisectors. The perpendicular bisectors of AA", BB", CC" bound a triangle A*B*C* called the

2nd Sharygin triangle. (Reference)A-vertex coordinates:

Trilinears-a^2+b*c : a*b-c^2 : a*c-b^2

## inner-Soddy

Given a triangle ABC with inner Soddy center S', the

inner Soddy triangleis the triangle A'B'C' formed by the points of tangency of the inner Soddy circle with the three mutually tangent circles centered at each of the vertices of ABC. (Reference)

Note: Given three noncollinear points, construct three tangent circles such that one is centered at each point and the circles are pairwise tangent to one another. Then there exist exactly two nonintersecting circles that are tangent to all three circles. These are called the inner and outer Soddy circles, and their centers are called the inner S and outer Soddy centers S^', respectively. (Reference)A-vertex coordinates:

Trilinears(a*(-a+b+c)+2*S)/(a*(-a+b+c)) : (b*(a+c-b)+S)/(b*(a-b+c)) : (c*(a+b-c)+S)/(c*(a-c+b))

## outer-Soddy

Given a triangle ABC with outer Soddy center S", the

outer Soddy triangleis the triangle A"B"C" formed by the points of tangency of the outer Soddy circle with the three mutually tangent circles centered at each of the vertices of ABC. (Reference)

Note: Given three noncollinear points, construct three tangent circles such that one is centered at each point and the circles are pairwise tangent to one another. Then there exist exactly two nonintersecting circles that are tangent to all three circles. These are called the inner and outer Soddy circles, and their centers are called the inner S and outer Soddy centers S^', respectively. (Reference)A-vertex coordinates:

Trilinears(a*(-a+b+c)-2*S)/(a*(-a+b+c)) : (b*(a+c-b)-S)/(b*(a-b+c)) : (c*(a+b-c)-S)/(c*(a-c+b))

## inner-squares

Build outwards ABC the square BCC

_{A}B_{A}(see figure). Lines AB_{A}and AC_{A}cut BC at A'_{B}and A'_{C}, respectively, and the perpendiculars through these points cut AB and AC at A"_{C}and A"_{B}, respectively. The quadrilateral A'_{B}A'_{C}A"_{B}A"_{C}is a square named theA-inner-inscribed-square(and also theA-Lucas square). The B- and C- inner-inscribed-squares can be built similarly. The centers of these squares are the vertices of theinner-squares triangles. (Reference)A-vertex coordinates:

Trilinears2*a : (a^2+b^2-c^2+2*S)/b : (a^2+c^2-b^2+2*S)/c

## outer-squares

Build inwards ABC the square BCC

_{A}B_{A}(see figure). Lines AB_{A}and AC_{A}cut BC at A'_{B}and A'_{C}, respectively, and the perpendiculars through these points cut AB and AC at A"_{C}and A"_{B}, respectively. The quadrilateral A'_{B}A'_{C}A"_{B}A"_{C}is a square named theA-outer-inscribed-square(and also theA-Lucas(-1) square). The B- and C- outer-inscribed-squares can be built similarly. The centers of these squares are the vertices of theouter-squares triangles. (Reference)A-vertex coordinates:

Trilinears2*a : (a^2+b^2-c^2-2*S)/b : (a^2+c^2-b^2-2*S)/c

## Stammler

The

Stammler triangleis the triangle formed by the centers of the Stammler circles. It is an equilateral triangle. (Reference)

Note: 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. (Reference)A-vertex coordinates:

Trilinearscos(A)-2*cos((B-C)/3) : cos(B)+2*cos(B/3+2*C/3) : cos(C)+2*cos(2*B/3+C/3)

## Steiner

The

Steiner triangleis the cevian triangle of the Steiner point X(99). (Reference)A-vertex coordinates:

Trilinears0 : -c*(a^2-b^2) : b*(a^2-c^2)

## submedial

Let ABC be a triangle, and suppose that C' is a point on side AB and that B' is a point on side AC. Let r(B'C') be the rectangle whose vertices are B', C', and the orthogonal projections of B' and C' onto side BC. Let R

_{A}be the rectangle r(B'C') of maximal area, which is obtained by taking A'B'C' to be the medial triangle of ABC. Let O_{A}be the center R_{A}, and define O_{B}and O_{C}cyclically. The central triangle O_{A}O_{B}O_{C}is named thesubmedial triangle. (Reference: X9813)A-vertex coordinates:

Trilinears2*a*b*c : (3*a^2+b^2-c^2)*c : (3*a^2+c^2-b^2)*b

## symmedial

The

symmedial triangleis the cevian triangle of the symmedian point X(6).A-vertex coordinates:

Trilinears0 : 1/c : 1/b

## T: ↑

## T(1,2)

See

Aquila triangle.

## T(-1,3)

See

excenters-incenter reflections triangle.

## T(-2,1)

See

excenters-incenter midpoints triangle.

## tangential

The

tangential triangleis the triangle enclosed by the lines tangent to the circumcircle of a given triangle ABC at its vertices. It is therefore the antipedal triangle of ABC with respect to the circumcenter O.A-vertex coordinates:

Trilinears-a : b : c

## (1st) tangential-midarc

The

tangential mid-arctriangle of a reference triangle ABC is the triangle A'B'C' whose sides are the tangents to the incircle at the intersections of the internal angle bisectors with the incircle, where the points of intersection nearest the vertices are chosen. (Reference)A-vertex coordinates:

Trilinears-2*b*c*sin(A/2) : c*(2*sin(B/2)*a+a+b-c) : b*(2*sin(C/2)*a+a+c-b)

## 2nd tangential-midarc

The

2nd tangential mid-arctriangle of a reference triangle ABC is the triangle A'B'C' whose sides are the tangents to the incircle at the intersections of the internal angle bisectors with the incircle, where the points of intersection furthest from the vertices are chosen. (Reference: Preamble above X8075)A-vertex coordinates:

Trilinears2*b*c*sin(A/2) : c*(2*sin(B/2)*a-(a+b-c)) : b*(2*sin(C/2)*a-(a+c-b))

## inner tri-equilateral

In a triangle ABC, let's inscribe three congruent equilateral triangles PA

_{b}A_{c}, PB_{c}B_{a}and PC_{a}C_{b}, with B_{a}, C_{a}on BC, C_{b}, A_{b}on CA and A_{c}, B_{c}on AB. There are two points P making possible this construction: P = P_{i}=X(15) and P = P_{o}=X(16). The equilateral triangles obtained in each case are here named theA-, B-, C- inner/outer equilateral triangles, respectively.

For the inner-equilateral triangles, theinner tri-equilateral triangleA_{i}B_{i}C_{i}is defined as the triangle bounded by the lines A_{b}A_{c}, B_{c}B_{a}and C_{a}C_{b}. (Reference: Preamble above X10631)A-vertex coordinates:

Trilinearsa*(SA-sqrt(3)*S)/(SA+sqrt(3)*S) : b : c

## outer tri-equilateral

In a triangle ABC, let's inscribe three congruent equilateral triangles PA

_{b}A_{c}, PB_{c}B_{a}and PC_{a}C_{b}, with B_{a}, C_{a}on BC, C_{b}, A_{b}on CA and A_{c}, B_{c}on AB. There are two points P making possible this construction: P = P_{i}=X(15) and P = P_{o}=X(16). The equilateral triangles obtained in each case are here named theA-, B-, C- inner/outer equilateral triangles, respectively.

For the outer-equilateral triangles, theouter tri-equilateral triangleA_{o}B_{o}C_{o}is defined as the triangle bounded by the lines A_{b}A_{c}, B_{c}B_{a}and C_{a}C_{b}. (Reference: Preamble above X10631)A-vertex coordinates:

Trilinearsa*(SA+sqrt(3)*S)/(SA-sqrt(3)*S) : b : c

## Trinh

There is a unique equilateral triangle AA

_{1}A_{2}inscribed in the circumcircle of triangle ABC, where, for concreteness, the labels are fixed so that AA_{1}A_{2}has the same orientation as ABC. Let BB_{1}B_{2}and CC_{1}C_{2}be the corresponding equilateral triangles. Let A' = B_{1}B_{2}∩C_{1}C_{2}, B' = C_{1}C_{2}∩A_{1}A_{2}and C' = A_{1}A_{2}∩B_{1}B_{2}. The triangle A'B'C' is named theTrinh triangleof ABC. (Reference: X7688)A-vertex coordinates:

Trilinears-a*(S^2+3*SA^2) : b*(S^2-3*SA*SB) : c*(S^2-3*SA*SC)

## U: ↑

## V: ↑

## inner-Vecten

Erect squares internally on each side of a triangle ABC. The centers of this squares are the vertices of the

inner-Vecten triangle.A-vertex coordinates:

Trilinears-a : (SC-S)/b : (SB-S)/c

## outer-Vecten

Erect squares externally on each side of a triangle ABC. The centers of this squares are the vertices of the

outer-Vecten triangle.A-vertex coordinates:

Trilinears-a : (SC+S)/b : (SB+S)/c

## W: ↑

## X: ↑

## X

_{3}-ABC reflectionsThis triangle is obtained by reflecting the circumcenter X

_{3}about A, B, C. Reference: TCCT 6.13, p. 160.A-vertex coordinates:

Trilinears-(3*S^2+SB*SC)/a : SB*b : SC*c

## X-parabola-tangential

Let A*B*C* be the vertex triangle of the medial and orthic triangles of ABC and A'B'C' the medial triangle of A*B*C*. Then A, B, C, A', B', C' lie on a parabola here named the

X-parabola of ABC. The tangents to this parabola at A,B,C bound theX-parabola-tangential triangle. (Reference: X12064)A-vertex coordinates:

Trilinears-(b^2-c^2)^2/a : (a^2-c^2)^2/b : (a^2-b^2)^2/c

## Y: ↑

## Yff central

Let three isoscelizers I

_{AC}I_{AB}, I_{BA}I_{BC}, and I_{CA}I_{CB}be constructed on a triangle ABC, one for each side. This makes all of the inner triangles similar to each other. However, there is a unique set of three isoscelizers for which the four interior triangles A'I_{BC}I_{CB}, I_{AC}B'I_{CA}, I_{AB}I_{BA}C', and A'B'C' are congruent. The innermost triangle A'B'C' is called theYff central triangle(Kimberling 1998, pp. 94-95). (See this figure).A-vertex coordinates:

Trilinears2*sin(A/2) : (2*a*sin(B/2)+a+b-c)/b : (2*a*sin(C/2)+a-b+c)/c

## Yff contact

The

Yff contact triangleis the cevian triangle of X(190).A-vertex coordinates:

Trilinears0 : -1/b/(a-c) : 1/c/(a-b)

## inner-Yff

The Yff circles are the two triplets of congruent circles in which each circle is tangent to two sides of a reference triangle (see Mathworld). The circles in each triplet have radius r

_{1}=r*R/(R+r) and r_{2}=r*R/(R-r), respectively. The centers of the circles in the first triplet are the vertices of theinner-Yff triangle. (Reference: Preamble just before X10037)A-vertex coordinates:

Trilinears-(a^4-2*(b^2+b*c+c^2)*a^2+(b^2-c^2)^2)/(2*a^2*b*c) : 1 : 1

## outer-Yff

The Yff circles are the two triplets of congruent circles in which each circle is tangent to two sides of a reference triangle (see Mathworld). The circles in each triplet have radius r

_{1}=r*R/(R+r) and r_{2}=r*R/(R-r), respectively. The centers of the circles in the second triplet are the vertices of theouter-Yff triangle. (Reference: Preamble just before X10037)A-vertex coordinates:

Trilinears(a^4-2*(b^2-b*c+c^2)*a^2+(b^2-c^2)^2)/(2*a^2*b*c) : 1 : 1

## inner-Yff tangents

Let L

_{A}be the line, other than BC, tangent to the B- and C-inner Yff circles. Define L_{B}, L_{C}cyclically. Let A' = L_{B}∩L_{C}and define B', C' cyclically. Triangle A'B'C' is theinner-Yff tangents triangle. (Reference: X10527)A-vertex coordinates:

Trilinears(b+c-a)*(a^3+(b+c)*a^2-(b^2+c^2)*a-(b^2-c^2)*(b-c))/(4*a^2*b*c) : 1 : 1

## outer-Yff tangents

Let M

_{A}be the line, other than BC, tangent to the B- and C-outer Yff circles. Define M_{B}, M_{C}cyclically. Let A" = M_{B}∩M_{C}and define B", C" cyclically. Triangle A"B"C" is theouter-Yff tangents triangle. (Reference: X10527)A-vertex coordinates:

Trilinears-(b+c-a)*(a^3+(b+c)*a^2-(b^2-4*b*c+c^2)*a-(b^2-c^2)*(b-c))/(4*a^2*b*c) : 1 : 1

## Yiu

The

Yiu triangleis the triangle formed by the centers of the Yiu circles. (Reference)

Note: The A- Yiu circle of a triangle ABC is the circle through A, the reflection of B in AC and the reflection of C in AB. (Reference)A-vertex coordinates:

Trilinears2*S*(SA^2-R^2*SA-S^2)/R : c*(S^2+SA*SC) : b*(S^2+SA*SB)

## Z: ↑

Return to ETC