Index of triangles

A:

ABC

The reference triangle

Equivalences:

anticomplement of (medial), complement of (anticomplementary).

(ABC-X3 reflections)-of-(ABC-X3 reflections), (anticomplementary)-of-(medial), (circumorthic)-of-(2nd circumperp), (2nd Euler)-of-(hexyl), (1st excosine)-of-(2nd Zaniah), (outer-Garcia)-of-(outer-Garcia), (Johnson)-of-(Johnson), (medial)-of-(anticomplementary), (orthic)-of-(excentral), (tangential)-of-(intouch).

Only for acute ABC: (1st circumperp)-of-(1st anti-circumperp), (2nd circumperp)-of-(circumorthic), (2nd Conway)-of-(2nd anti-Conway), (3rd Euler)-of-(3rd anti-Euler), (4th Euler)-of-(4th anti-Euler), (excentral)-of-(orthic), (hexyl)-of-(2nd Euler), (6th mixtilinear)-of-(6th anti-mixtilinear), (Wasat)-of-(anti-Wasat).

A-vertex coordinates:

Trilinears 1 : 0 : 0

AAOA (antiAOA)

Let L'_A be the orthic axis of the A-anti-altimedial triangle, and define L'_B, L'_C cyclically. Let A" = L'_B∩L'_C, and define B", C" cyclically. The triangle A"B"C" is the antialtimedial orthic axes triangle, or AAOA triangle for short. (Reference: preamble just before X15015)

A-vertex coordinates:

Trilinears
(a^8+3*a^4*b^2*c^2-2*(b^2+c^2)*a^6+2*(b^4-c^4)*(b^2-c^2)*a^2-(b^4-c^4)^2)/(a*b*c) :
(-b^2*(a^2-b^2)-c^4+a^4)*c :
(-c^2*(a^2-c^2)-b^4+a^4)*b

ABC-X₃ reflections

This triangle is obtained by reflecting A, B, C about the circumcenter X₃. Reference: TCCT 6.12, p. 159.

Equivalences:

isogonal of (infinite-altitude), anticomplement of (Euler).

(1st anti-circumperp)-of-(2nd circumperp), (2nd anti-circumperp-tangential)-of-(anti-Mandart-incircle), (circumorthic)-of-(1st circumperp), (Euler)-of-(anticomplementary), (2nd Euler)-of-(excentral), (Mandart-incircle)-of-(2nd circumperp tangential), (medial)-of-(anti-Euler), (orthic)-of-(hexyl).

Only for acute ABC: (1st circumperp)-of-(circumorthic), (2nd circumperp)-of-(1st anti-circumperp), (3rd Euler)-of-(4th anti-Euler), (4th Euler)-of-(3rd anti-Euler), (Hutson intouch)-of-(tangential).

A-vertex coordinates:

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

AaBbCc and (AaBbCc)*

See 2nd and 4th Przybyłowski-Bollin triangle.

Aguilera

In a scalene and non-rectangle triangle ABC, let MaMbMc be its medial triangle. Let ωa be the circle centered at Ma and passing through B and C, and define ωb and ωc cyclically. Inside ABC, let Oa be center of the circle, closest to A, which is externally tangent to ωa and is inscribed in angle BAC. Define Ob and Oc cyclically. The triangle OaObOc is called the (1st) Aguilera triangle of ABC. (Reference: Preamble before X60877).

A-vertex coordinates:

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

Aguilera-Pavlov

Let A'B'C' be the (1st) Aguilera triangle of ABC and A", B", C" the inverses of A', B', C' in the incircle of ABC, respectively. The triangle A"B"C" is called the (1st) Aguilera-Pavlov triangle of ABC. (Reference: Preamble before X60877).

A-vertex coordinates:

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

AiBiCi and (AiBiCi)*

See 1st and 3rd Przybyłowski-Bollin triangle.

altimedial

Let M_A, M_B, M_C be the midpoints of sides BC, CA, AB, respectively, and H_A, H_B, H_C the feet of the altitudes from A, B, C, respectively. The triangles H_AM_CM_B, M_CH_BM_A, M_BM_AH_C are the A-, B- and C-altimedial triangles. (References: Hyacinthos 253 and preamble just before X15015)

A-altimedial triangle trilinear vertex coordinates:

H_A 0 : (-c^2+a^2+b^2)*c : (-b^2+c^2+a^2)*b
M_C b : a : 0
M_B c : 0 : a

Note: Let P be a triangle center. Let P_A be P-of-A-altimedial-triangle, and define P_B, P_C cyclically. Triangle P_AP_BP_C is the P-of-altimedial-triangles triangle, or P-altimedial triangle for short.

altimedial orthic axes (AOA)

Let L_A be the orthic axis of the A-altimedial triangle, and define L_B, L_C cyclically. Let A' = L_B∩L_C, and define B', C' cyclically. The triangle A'B'C' is the altimedial orthic axes triangle, or AOA triangle for short. (Reference: preamble just before X15015)

A-vertex coordinates:

Trilinears
((b^2+c^2)*a^6-(b^4+c^4)*a^4-(b^4-c^4)*(b^2-c^2)*a^2+(b^4-c^4)^2)/a :
(b^8-3*b^6*a^2-(3*c^4-4*c^2*a^2-a^4)*b^4+(c^2-a^2)*(7*c^2-3*a^2)*b^2*a^2+2*(c^4-a^4)*(c^2-a^2)^2)/b :
(c^8-3*c^6*a^2-(3*b^4-4*b^2*a^2-a^4)*c^4+(b^2-a^2)*(7*b^2-3*a^2)*c^2*a^2+2*(b^4-a^4)*(b^2-a^2)^2)/c

1st and 2nd Altintas-isodynamic

Let ABC be a triangle. The parallel line to BC from P = X(15) (1st isodynamic point) intersect the circumcircle of ABC in A₁, A₂ and the Simson lines of A₁, A₂ cut at A'. Define B', C' cyclically. Then A'B'C' is equilateral and it is the 1st Altintas-isodynamic triangle. (Reference: Preamble just before X40933)

When P=X(16) (2nd isodynamic point), the parallel line to BC from P intersect the circumcircle of ABC in two imaginary points, but the Simson lines of these points still intersect in a real point A". The triangle A"B"C" built in this way is also equilateral and is the 2nd Altintas-isodynamic triangle.

A-vertex coordinates:

1st Altintas-isodynamic triangle:

Trilinears -((S^2+SB*SC)*sqrt(3)*S+(SA+3*SW)*S^2-2*(S^2-SA*SW+SW^2)*SA)/(a*(S^2-sqrt(3)*SA*S-2*SA*SW)) : SC/b : SB/c
2nd Altintas-isodynamic triangle:

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

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

adjunct anti-altimedial

Let A_AB_AC_A, A_BB_BC_B, A_CB_CC_C be the A-, B- and C- anti-altimedial triangles of ABC. The triangles A_AA_BA_C, B_AB_BB_C, C_AC_BC_C are the A-, B- and C- adjunct anti-altimedial triangles, respectively. (Reference: preamble just before X15015)

A-adjunct anti-altimedial triangle trilinear vertex coordinates:

A_A -a : (a^2+b^2-c^2)/b : (a^2-b^2+c^2)/c
A_B 1/a : 1/(-a^2+b^2+c^2)*b : -1/c
A_C 1/a : -1/b : 1/(-a^2+b^2+c^2)*c

Note: Let P be a triangle center. Let P"_A be P-of-A-adjunct-anti-altimedial-triangle, and define P"_B, P"_C cyclically. Triangle P"_AP"_BP"_C is the P-of-adjunct-anti-altimedial-triangles triangle, or P-adjunct-anti-altimedial triangle for short.

anticomplementary

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

Equivalences:

isogonal of (tangential), complement of (Gemini 111), anticomplement of (ABC).

(anti-Euler)-of-(ABC-X3 reflections), (3rd anti-Euler)-of-(1st circumperp), (4th anti-Euler)-of-(2nd circumperp), (anti-Wasat)-of-(excentral), (Aquila)-of-(outer-Garcia), (1st excosine)-of-(intouch), (Johnson)-of-(anti-Euler), (medial)-of-(Gemini 111), (orthic)-of-(2nd Conway), (tangential)-of-(inner-Conway), (X3-ABC reflections)-of-(Johnson).

Only for acute ABC: (excentral)-of-(1st anti-circumperp), (hexyl)-of-(circumorthic), (6th mixtilinear)-of-(orthic), (Ursa-minor)-of-(tangential), (Wasat)-of-(3rd anti-Euler).

A-vertex coordinates:

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

anti-triangles

If T is a triangle of ABC then the anti-T triangle is the triangle whose T-triangle is ABC.

anti-altimedial

The triangle A_AB_AC_A whose A-altimedial triangle is ABC. (Reference: preamble just before X15015)

A-anti-altimedial triangle trilinear vertex coordinates:

A_A -a : (a^2+b^2-c^2)/b : (a^2-b^2+c^2)/c
B_A a/(a^2-b^2+c^2) : 1/b : -1/c
C_A a/(a^2+b^2-c^2) : -1/b : 1/c

Note: Let P be a triangle center. Let P'_A be P-of-A-anti-altimedial-triangle, and define P'_B, P'_C cyclically. Triangle P'_AP'_BP'_C is the P-of-anti-altimedial-triangles triangle, or P-anti-altimedial triangle for short.

anti-Aquila

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

Equivalences:

complement of (outer-Garcia).

(6th anti-mixtilinear)-of-(hexyl), (outer-Garcia)-of-(medial), (Kosnita)-of-(intouch), (orthic)-of-(2nd circumperp), (tangential)-of-(incircle-circles), (Trinh)-of-(Hutson intouch).

A-vertex coordinates:

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

anti-Ara

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

Equivalences:

(Hutson intouch)-of-(anti-excenters-reflections).

Only for acute ABC: (incircle-circles)-of-(circumorthic), (intouch)-of-(orthic).

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.

Equivalences:

(anticomplementary)-of-(2nd anti-extouch).

A-vertex coordinates:

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

anti-Atik

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

Notes: Only when ABC is acute. A' = AX(69) ∩ perpendicular(X(18909), BC)

A-vertex coordinates:

Trilinears S^2*(4*R^2-SA)/(a*SA^2) : SB/b : SC/c

anti-1st/2nd Auriga

Triangles A'B'C' whose 1st/2nd Auriga triangle is ABC..

A-vertex coordinates:

anti-1st Auriga:

Trilinears -(D*a+(a+b+c)*(a*(b^2+c^2)-(b+c)*(b-c)^2))/a : b*(a+b+c)*(a-b+c)-D : c*(a+b+c)*(a+b-c)-D
anti-2nd Auriga:

Trilinears -(-D*a+(a+b+c)*(a*(b^2+c^2)-(b+c)*(b-c)^2))/a : b*(a+b+c)*(a-b+c)+D : c*(a+b+c)*(a+b-c)+D
where where D=sqrt(r*R+4*R^2).

1st anti-Brocard

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

Note: For construction, see Bernard Gibert's anti-Brocard triangles and related topics.

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.

Note: For construction, see Bernard Gibert's anti-Brocard triangles and related topics.

A-vertex coordinates:

Trilinears a*((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.

Note: For construction, see Bernard Gibert's anti-Brocard triangles and related topics.

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.

Note: For construction, see Bernard Gibert's anti-Brocard triangles and related topics.

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.

Equivalences:

anticomplement of (orthic).

(ABC-X3 reflections)-of-(circumorthic), (6th anti-mixtilinear)-of-(Gemini 111), (circumorthic)-of-(ABC-X3 reflections), (Euler)-of-(4th anti-Euler), (2nd Euler)-of-(anti-Euler), (medial)-of-(3rd anti-Euler), (orthic)-of-(anticomplementary).

Only for acute ABC: (2nd anti-circumperp-tangential)-of-(anti-Hutson intouch), (Mandart-incircle)-of-(tangential).

A-vertex coordinates:

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

1st anti-circumperp-tangential

The triangle A'B'C' whose 1st circumperp-tangential triangle is ABC. It is the Anti-Mandart-incircle triangle.

2nd anti-circumperp-tangential

The triangle A'B'C' whose 2nd circumperp-tangential is ABC.

Equivalences:

(ABC-X3 reflections)-of-(Mandart-incircle), (1st anti-circumperp)-of-(Hutson intouch), (circumorthic)-of-(intouch), (2nd Euler)-of-(Ursa-minor), (Mandart-incircle)-of-(5th mixtilinear).

Only for acute ABC: (Hutson intouch)-of-(intangents).

Notes:

1) For any ABC. A' = AX(56) ∩ perpendicular(X(7354), BC)

2) Continuing with the construction 2) of the anti-tangential-midarc triangle, the line A_bA_c is tangent to the incircle of ABC in a point A'; B', C' are defined similarly. Then A'B'C' is the 2nd anti-circumperp-tangential triangle. (César Lozada, Dec. 8, 2021).

A-vertex coordinates:

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

(1st) anti-Conway

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

Equivalences:

(medial)-of-(2nd anti-extouch).

Note: The A-vertex of the anti-Conway triangle is the A' = {perpendicular to BC through X(578)} ∩ AX(6).

A-vertex coordinates:

Trilinears a^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.

Equivalences:

(medial)-of-(orthic).

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-Ehrmann-mid

The triangle A'B'C' whose Ehrmann-mid triangle is ABC.

Equivalences:

(anti-Euler)-of-(Johnson), (Johnson)-of-(X3-ABC reflections).

Only for acute ABC: (excentral)-of-(Ehrmann-side).

A-vertex coordinates:

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

anti-Euler

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

Equivalences:

anticomplement of (Johnson).

(anticomplementary)-of-(ABC-X3 reflections), (anti-Ehrmann-mid)-of-(Johnson), (3rd anti-Euler)-of-(2nd circumperp), (4th anti-Euler)-of-(1st circumperp), (anti-Wasat)-of-(hexyl), (anti-X3-ABC reflections)-of-(Gemini 111), (Johnson)-of-(anticomplementary).

Only for acute ABC: (excentral)-of-(circumorthic), (hexyl)-of-(1st anti-circumperp), (Ursa-minor)-of-(anti-Hutson intouch), (Wasat)-of-(4th anti-Euler).

Note: A', B', C' are the reflections of the orthocenter of ABC in A, B, C, respectively.

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.

Equivalences:

anticomplement of (anti-Wasat).

(anticomplementary)-of-(1st anti-circumperp), (anti-Euler)-of-(circumorthic), (4th anti-Euler)-of-(ABC-X3 reflections), (anti-Wasat)-of-(anticomplementary), (Johnson)-of-(4th anti-Euler).

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.

Equivalences:

(anticomplementary)-of-(circumorthic), (anti-Euler)-of-(1st anti-circumperp), (3rd anti-Euler)-of-(ABC-X3 reflections), (anti-Wasat)-of-(anti-Euler), (Johnson)-of-(3rd anti-Euler).

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.

Equivalences:

(anti-Hutson intouch)-of-(anti-Ara).

Only for acute ABC: (5th mixtilinear)-of-(orthic).

A-vertex coordinates:

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

anti-excenters-reflections

See anti-excenters-incenter reflections.

2nd anti-extouch

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

Equivalences:

(anticomplementary)-of-(anti-Conway), (medial)-of-(anti-Ascella).

Note: Only when ABC is acute. A' = AX(3) ∩ perpendicular(X(1181), BC)

A-vertex coordinates:

Trilinears S^2*a/SA : SB*b : SC*c

anti-inner-Garcia

The triangle A'B'C' whose inner-Garcia triangle is ABC.

Note: If A"B"C" is the inner-Garcia triangle of ABC, then A' = A"X(3) ∩ perpendicular(X(6326), B"C").

A-vertex coordinates:

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

anti-inner-Grebe

The triangle A'B'C' whose inner-Grebe triangle is ABC.

Notes: For any ABC. A' = AX(6) ∩ perpendicular(X(1588), BC)

A-vertex coordinates:

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

anti-outer-Grebe

The triangle A'B'C' whose outer-Grebe triangle is ABC.

Notes: For any ABC. A' = AX(6) ∩ perpendicular(X(1587), BC)

A-vertex coordinates:

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

anti-Honsberger

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

Notes: Only for ABC acute. A' = AX(6) ∩ perpendicular(X(182), BC)

A-vertex coordinates:

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

anti-Hutson intouch

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

Equivalences:

(anti-Ursa minor)-of-(anti-Euler), (tangential)-of-(ABC-X3 reflections), (Trinh)-of-(X3-ABC reflections).

Only for acute ABC: (anti-Mandart-incircle)-of-(circumorthic), (Aquila)-of-(Trinh), (2nd circumperp tangential)-of-(1st anti-circumperp).

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.

Equivalences:

(anti-inverse-in-incircle)-of-(Johnson), (circumorthic)-of-(Ara), (tangential)-of-(X3-ABC reflections).

Only for acute ABC: (Aquila)-of-(tangential).

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.

Equivalences:

anticomplement of (tangential).

(anti-incircle-circles)-of-(Johnson), (anti-Ursa minor)-of-(Gemini 111), (tangential)-of-(anticomplementary).

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- 1st/2nd Kenmotu-centers

The triangle A'B'C' whose 1st/2nd Kenmotu-centers triangle is ABC.

A-vertex coordinates:

anti-1st Kenmotu-centers:

Trilinears -(b^2+c^2+2*S)/a : b : c
anti-2nd Kenmotu-centers:

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

anti-1st/2nd Kenmotu-free-vertices

The triangle A'B'C' whose 1st/2nd Kenmotu-free-vertices triangle is ABC.

A-vertex coordinates:

anti-1st Kenmotu-free-vertices:

Trilinears (SA+S)*(SB+SC+2*S)/a : (SB-S)*b : (SC-S)*c
anti-2nd Kenmotu-free-vertices:

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

anti-Lucas(-1) homothetic and anti-Lucas(+1) homothetic

The triangle A'B'C' whose Lucas(-1) homothetic/Lucas(+1) homothetic triangle is ABC.

A-vertex coordinates:

anti-Lucas(-1) homothetic:

Trilinears S*(2*S^2-(SA+SW)*S+SA^2)/a : b*(2*S^2-(SB+SW)*S+SB^2) : c*(2*S^2-(SC+SW)*S+SC^2)
anti-Lucas(+1) homothetic:

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

anti-Mandart-incircle

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

Equivalences:

(anti-Hutson intouch)-of-(2nd circumperp), (anti-Ursa minor)-of-(excentral), (2nd circumperp tangential)-of-(ABC-X3 reflections), (tangential)-of-(1st circumperp).

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.

Equivalences:

anticomplement of (submedial), complement of (orthic).

(1st anti-circumperp)-of-(Gemini 110), (2nd Euler)-of-(anti-X3-ABC reflections), (orthic)-of-(medial), (submedial)-of-(anticomplementary).

Only for acute ABC: (anti-Aquila)-of-(2nd Euler).

A-vertex coordinates:

Trilinears 2*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.

Equivalences:

4th anti-Brocard-of-circumsymmedial.

A-vertex coordinates:

Trilinears a*((-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))

1st anti-orthosymmedial

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

Notes: Only for ABC acute. A' = AX(1297) ∩ perpendicular(X(19158), BC)

A-vertex coordinates:

Trilinears
(S^4+3*SA^2*S^2+2*(SA^2-SB*SC-SW^2)*SA^2)*a/(2*SA+SB+SC) : ((SA-SB)*S^2-2*(SA*SC-SB^2)*SA)*b : ((SA-SC)*S^2-2*(SA*SB-SC^2)*SA)*c

anti-1st Parry

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

A-vertex coordinates:

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

anti-2nd Parry

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

A-vertex coordinates:

Trilinears -(2*S^4+(5*SA^2-8*SB*SC+SW^2)*S^2+3*(SA^2-4*SB*SC-SW^2)*SA^2)/(a*(3*SA-SW)) : b*(S^2-3*SA*SB) : c*(S^2-3*SA*SC)

anti-reflection

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

A-vertex coordinates:

Trilinears Coordinates not found yet

1st anti-Sharygin

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

Notes: Only for ABC acute. A' = AX(8795) ∩ perpendicular(X(8884), BC)

A-vertex coordinates:

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

anti-tangential-midarc

The triangle A'B'C' whose tangential-midarc triangle is ABC.

Equivalences:

(anti-Hutson intouch)-of-(Mandart-incircle), (tangential)-of-(2nd anti-circumperp-tangential).

Notes:

1) A' = AX(65) ∩ perpendicular(X(1), BC), when ABC is acute.

2) Let a_b, a_c be the circles both tangent to AI at I=X(1) and passing the first through B and the second through C. Let A_b be the point, other than B, at which a_b cuts AB, and A_c the point, other than C, at which a_c cuts AC. Define {B_c, B_a}, {C_a, C_b} cyclically. Then the triangle bounded by lines A_bA_c, B_cB_a, C_aC_b is the anti-tangential-midarc triangle of ABC. (César Lozada, Dec. 8, 2021)

A-vertex coordinates:

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

3rd anti-tri-squares

The triangle A'B'C' whose 3rd tri-squares triangle is ABC.

A' = AX(1328) ∩ perpendicular(X(486), BC) (Only for ABC acute)

Build squares BCC_aB_a, CAA_bC_b and ABB_cA_c inwards ABC. A' = A_bB_a ∩ A_cC_a.

A-vertex coordinates:

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

4th anti-tri-squares

The triangle A'B'C' whose 4th tri-squares triangle is ABC.

A' = AX(1327) ∩ perpendicular(X(485), BC) (Only for ABC acute)

Build squares BCC_aB_a, CAA_bC_b and ABB_cA_c outwards ABC. A' = A_bB_a ∩ A_cC_a.

A-vertex coordinates:

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

anti-3rd/4th tri-squares-central

The triangle A'B'C' whose 3rd/4th tri-squares-central triangle is ABC.

A-vertex coordinates:

anti-3rd tri-squares-central:

Trilinears -(b^2+c^2+3*S)/a : (b^2+S)/b : (c^2+S)/c
anti-4th tri-squares-central:

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

anti-Ursa minor

The triangle A'B'C' whose Ursa minor triangle is ABC.

Equivalences:

complement of (tangential).

(anti-Hutson intouch)-of-(Euler), (anti-inverse-in-incircle)-of-(Gemini 110), (tangential)-of-(medial).

Only for acute ABC: (anti-Mandart-incircle)-of-(orthic), (2nd circumperp tangential)-of-(2nd Euler).

Construction: Let A₀B₀C₀ be the orthic triangle of ABC. Let A₁ be the intersection of the parallel to A₀B₀ through B and the parallel to A₀C₀ through C. Let A₂ be the midpoint of A and A₁. Build B₂, C₂ cyclically. Then A₂B₂C₂ is the anti-Ursa minor triangle of ABC. (Added Nov. 14, 2020)

A-vertex coordinates:

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

anti-Wasat

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

Equivalences:

complement of (3rd anti-Euler).

(anticomplementary)-of-(orthic), (anti-Euler)-of-(2nd Euler), (3rd anti-Euler)-of-(medial), (4th anti-Euler)-of-(Euler), (1st excosine)-of-(anti-Ara).

A-vertex coordinates:

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

anti-X3-ABC reflections

The triangle A'B'C' whose X3-ABC reflections triangle is ABC.

Equivalences:

complement of (Johnson).

(anti-Euler)-of-(Gemini 110), (Ehrmann-mid)-of-(Johnson), (Johnson)-of-(medial).

Only for acute ABC: (hexyl)-of-(6th anti-mixtilinear), (Hutson intouch)-of-(Trinh), (incircle-circles)-of-(tangential).

A-vertex coordinates:

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

anti-inner/outer Yff

The triangle A'B'C' whose inner-/outer- Yff triangle is ABC.

A-vertex coordinates:

anti-inner-Yff:

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

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

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 triangle of ABC. (Reference: X5574)

A-vertex coordinates:

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

AOA

See altimedial orthic axes triangle.

Apollonius

The Apollonious circle is the circle internally tangent to all the excircles (Reference: MathWorld -- Apollonius Circle.). 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)

Equivalences:

isogonal of (2nd Conway).

A-vertex coordinates:

Trilinears a/(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.

Equivalences:

(anticomplementary)-of-(outer-Garcia), (anti-incircle-circles)-of-(intouch), (circumorthic)-of-(excentral), (Ehrmann-side)-of-(hexyl), (2nd Euler)-of-(6th mixtilinear).

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)

Equivalences:

Only for acute ABC: (excentral)-of-(tangential), (Wasat)-of-(1st excosine).

A-vertex coordinates:

Trilinears a*(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 the Ascella triangle. (Reference: Preamble just before X8726)

Equivalences:

complement of (2nd extouch).

A-vertex coordinates:

Trilinears 2*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 v² = |UV|² - u² = 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 the Atik triangle. (Reference: Preamble just before X8580))

The Atik triangle is also the triangle bounded by the polars of the incenter X(1) with respect to the excircles (Sept 10, 2017).

Let I, I_a be the incenter and A-excenter of ABC. Let a' be the radical axis of the A-excircle and the circle with diameter II_a and denote b', c' cyclically. The triangle bounded by these radical axes is the Atik triangle (July 7, 2020).

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:

Trilinears (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

where D=sqrt(r*R+4*R^2)

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:

Trilinears (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

where D=sqrt(r*R+4*R^2)

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.) The Ayme triangle is T_AT_BT_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:

Bankoff equilateral

Let ABC be a triangle with circumcenter O. Centering at O, let A_b be the rotation of A toward B by an angle |π/6| and let A_c be the rotation of A toward C by the same angle |π/6| (triangle OA_bA_c is an equilateral triangle). Build (B_c, B_a) and (C_a, C_b) cyclically and let A_m, B_m, C_m be the midpoints of B_cC_b, C_aA_c, A_bB_a, respectively. The triangle A_mB_mC_m is equilateral and is named the Bankoff equilateral triangle of ABC. (Reference: X34551)

A-vertex coordinates:

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

BCE

Let E_a, E_b, E_c be the excenters of ABC and J_a, J_b, J_c the E_a-, E_b-, E_c- excenters of E_aBC, E_bCA and E_cAB, respectively. The triangle J_aJ_bJ_c is the BCE triangle of ABC.

A-vertex coordinates:

Trilinears -1 : 1-2*sin(C/2) : 1-2*sin(B/2)

BCE-incenters

Let E_a, E_b, E_c be the excenters of ABC and I_a, I_b, I_c the incenters of E_aBC, E_bCA and E_cAB, respectively. The triangle I_aI_bI_c is the BCE-incenters triangle of ABC.

A-vertex coordinates:

Trilinears -1 : 1+2*sin(C/2) : 1+2*sin(B/2)

BCI

Let I be the incenter of ABC and I_a, I_b, I_c the incenters of IBC, ICA and IAB, respectively. The triangle I_aI_bI_c is the BCI triangle of ABC. (Reference: MathWorld -- BCI Triangle.)

A-vertex coordinates:

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

BCI-excenters

Let I be the incenter of ABC and E_a, E_b, E_c the I- excenters of IBC, ICA and IAB, respectively. The triangle E_aE_bE_c is the BCI-excenters triangle of ABC.

A-vertex coordinates:

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

Bevan-antipodal

Let V be the Bevan point X(40) of ABC and V_a, V_b, V_c the antipodes of V on the circumcircles of VBC, VCA and VAB, respectively. The triangle V_aV_bV_c is the Bevan-antipodal triangle of ABC. (Reference: X223)

Note: The Bevan-antipodal triangle is the anticevian triangle of X(57).

A-vertex coordinates:

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

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 triangle of ABC. (Reference: MathWorld -- Brocard Triangles.)

Equivalences:

isogonal of (3rd Brocard).

Only for acute ABC: (1st circumperp)-of-(7th Brocard), (2nd circumperp)-of-(10th Brocard).

A-vertex coordinates:

Trilinears a*b*c : c^3 : b^3

1st Brocard-reflected

The triangles whose vertices are the reflections of the vertices of the 1st Brocard triangle in the sidelines of ABC is the 1st Brocard-reflected triangle of ABC. (Reference: CTC - Table 32)

A-vertex coordinates:

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

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 the 2nd Brocard triangle of ABC. (Reference: MathWorld -- Brocard Triangles.)

Equivalences:

isogonal of (4th Brocard).

A-vertex coordinates:

Trilinears b^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 triangle of ABC. (Reference: MathWorld -- Brocard Triangles.)

Equivalences:

isogonal of (1st Brocard).

A-vertex coordinates:

Trilinears b^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 triangle of ABC. (Reference: MathWorld -- Brocard Triangles.)
In a triangle ABC, let a_b be the circle through A and tangent to BC at B and let a_c be the circle through A and tangent to BC at C. Let A* be the point of intersection, other than A, of a_b and a_c. Define B*, C* cyclically. Then A*B*C* is the 4th Brocard triangle of ABC. (Apr. 15, 2024)
Equivalences:

isogonal of (2nd Brocard).

A-vertex coordinates:

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

5th Brocard

The 5th Brocard triangle is 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

7th Brocard

In the plane of a triangle ABC, let O = X(3), the circumcenter. Let A' be the point, other than O, in which the line AO meets the Brocard circle, and define B' and C' cyclically. The triangle A'B'C', is the 7th Brocard triangle. (Reference: Dan Reznik and Peter Moses, preamble just before X39643)

Equivalences:

(ABC-X3 reflections)-of-(10th Brocard), (1st anti-circumperp)-of-(1st Brocard).

A-vertex coordinates:

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

8th Brocard

Let A'B'C' be the 7th Brocard triangle of ABC and A", B", C" the circumcircle-inverses of A', B', C', respectively. The triangle A"B"C", is the 8th Brocard triangle. (Reference: Preamble just before X39642)

Note: A", B", C" are collinear on the Brocard axis of ABC.

Equivalences:

isogonal of (9th Brocard).

A-vertex coordinates:

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

9th Brocard

Let A"B"C" be the 8th Brocard triangle of ABC and A*, B*, C* the isogonal conjugates of A", B", C", respectively. The triangle A*B*C*, is the 9th Brocard triangle. (Reference: Preamble just before X39642)

Equivalences:

isogonal of (8th Brocard).

A-vertex coordinates:

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

10th Brocard

The reflection of 7th Brocard triangle in the center X(182) of Brocard circle is the 10th Brocard triangle. (Reference: Preamble just before X43803)

Equivalences:

(ABC-X3 reflections)-of-(7th Brocard), (circumorthic)-of-(1st Brocard).

A-vertex coordinates:

Trilinears -(a^8-2*(b^4+c^4)*a^4+2*(b^4-c^4)*(b^2-c^2)*a^2-(b^2-c^2)^4)/a :
b*(a^6-(b^2-3*c^2)*a^4+(b^4-c^4)*a^2-(b^2-c^2)^3) :
c*(a^6+(3*b^2-c^2)*a^4-(b^4-c^4)*a^2+(b^2-c^2)^3)

11th Brocard

Let A' be the Psi-transform of the antipode of A in the circumcircle of ABC, and define B' and C' cyclically. The triangle A'B'C' is the 11th Brocard triangle of ABC. It inscribed in the Brocard circle. (Reference: Bernard Gibert's Brocard triangles.)

A'B'C' is the circumcevian triangle of X(999) wrt to the 10th-Brocard triangle. (X(999)-of-10th-Brocard triangle=X(9306)-of-ABC). The Psi-transform or Psi-image of a point is defined in the preamble just before X(14649).

A-vertex coordinates:

Trilinears (a^6+(b^2+c^2)*a^4-3*(b^2-c^2)^2*a^2+(b^4-c^4)*(b^2-c^2))/a :
(a^4-2*(2*b^2-3*c^2)*a^2+(b^2-c^2)*(3*b^2-c^2))*b :
(a^4+2*(3*b^2-2*c^2)*a^2+(b^2-c^2)*(b^2-3*c^2))*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.

Equivalences:

anticomplement of (5th Euler).

A-vertex coordinates:

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

circumnormal

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

Equivalences:

(ABC-X3 reflections)-of-(circumtangential), (anti-Aquila)-of-(Stammler), (2nd anti-circumperp-tangential)-of-(Stammler), (anti-X3-ABC reflections)-of-(Stammler), (circumorthic)-of-(circumtangential), (Euler)-of-(Stammler), (2nd Euler)-of-(Stammler), (4th Euler)-of-(Stammler), (outer-Garcia)-of-(circumtangential), (Hutson intouch)-of-(Stammler), (Johnson)-of-(circumtangential), (Kosnita)-of-(circumtangential).

A-vertex coordinates:

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

circumorthic

The circumcevian triangle of the orthocenter of ABC is its circumorthic triangle.

Equivalences:

anticomplement of (2nd Euler).

(ABC-X3 reflections)-of-(1st anti-circumperp), (1st anti-circumperp)-of-(ABC-X3 reflections), (anti-incircle-circles)-of-(anti-Ara), (Ehrmann-side)-of-(Johnson), (Euler)-of-(3rd anti-Euler), (2nd Euler)-of-(anticomplementary), (medial)-of-(4th anti-Euler), (orthic)-of-(anti-Euler).

Only for acute ABC: (2nd anti-circumperp-tangential)-of-(tangential), (Aquila)-of-(orthic), (Mandart-incircle)-of-(anti-Hutson intouch).

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).

Equivalences:

anticomplement of (3rd Euler).

(ABC-X3 reflections)-of-(2nd circumperp), (2nd circumperp)-of-(ABC-X3 reflections), (Euler)-of-(hexyl), (3rd Euler)-of-(anticomplementary), (4th Euler)-of-(anti-Euler), (Hutson intouch)-of-(2nd circumperp tangential), (medial)-of-(excentral).

A-vertex coordinates:

Trilinears a : c-b : b-c

1st circumperp tangential

The tangential triangle of the 1st circumperp triangle. (TCCT, 6-23, p. 164).

See anti-Mandart-incircle triangle.

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).

Equivalences:

anticomplement of (4th Euler).

(ABC-X3 reflections)-of-(1st circumperp), (Ara)-of-(incircle-circles), (1st circumperp)-of-(ABC-X3 reflections), (Euler)-of-(excentral), (3rd Euler)-of-(anti-Euler), (4th Euler)-of-(anticomplementary), (excentral)-of-(anti-Aquila), (Hutson intouch)-of-(anti-Mandart-incircle), (medial)-of-(hexyl).

A-vertex coordinates:

Trilinears -a : b+c : b+c

2nd circumperp tangential

The tangential triangle of the 1st circumperp triangle. (TCCT, 6-24, p. 165).

Equivalences:

(anti-Hutson intouch)-of-(1st circumperp), (anti-Mandart-incircle)-of-(ABC-X3 reflections), (anti-Ursa minor)-of-(hexyl), (tangential)-of-(2nd circumperp).

A-vertex coordinates:

Trilinears b*c+(a+b)*(a+c) : -b*(a+b-c) : -c*(a-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 triangle has 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.

Equivalences:

(ABC-X3 reflections)-of-(circumnormal), (circumorthic)-of-(circumnormal), (Ehrmann-mid)-of-(Stammler), (3rd Euler)-of-(Stammler), (outer-Garcia)-of-(circumnormal), (Johnson)-of-(circumnormal), (Kosnita)-of-(circumnormal), (Mandart-incircle)-of-(Stammler), (medial)-of-(Stammler).

A-vertex coordinates:

Trilinears -1 : sin((B-C)/3)*csc((B+2*C)/3) : sin((C-B)/3)*csc((C+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. The Conway triangle is the triangle bounded by the lines A_bA_c, B_cB_a and C_aC_b. (Reference: X7411)

Note: See this construction at MathWorld -- Conway Circle..

Equivalences:

anticomplement of (2nd extouch).

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 A₂ = A_bB_a∩A_cC_a, and define B₂ and C₂ cyclically. The triangle A₂B₂C₂ is the 2nd Conway triangle of ABC. (Reference: Preamble just before X9776)

Equivalences:

isogonal of (Apus), anticomplement of (excentral).

(Ara)-of-(inner-Conway), (excentral)-of-(anticomplementary), (Wasat)-of-(Gemini 111).

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 A₃ be the intersection of the tangents to the Conway circle at B_a and C_a, and define B₃, C₃ cyclically. The triangle A₃B₃C₃ is the 3rd 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.

Equivalences:

(anticomplementary)-of-(inverse-in-Conway).

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 A₄ be the intersection of the tangents to the Conway circle at A_b and A_c, and define B₄, C₄ cyclically. The triangle A₄B₄C₄ is the 4th 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 A₅ be the intersection of the tangents to the Conway circle at B_c and C_b, and define B₅, C₅ cyclically. Triangle A₅B₅C₅ is named the 5th 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. The inner-Conway triangle is the triangle bounded by the lines A_bA_c, B_cB_a and C_aC_b. (Reference: Preamble just before X11677)

Equivalences:

anticomplement of (intouch).

(anti-Ara)-of-(2nd Conway).

Notes: See this construction at MathWorld -- 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.

Dao

Let (a) denote the ellipse through A with foci B and C and define (b) and (c) cyclically. Let d_a be the line joining the intersection points of of (b) and (c) and denote A'= d_a ∩ BC. Build B' and C' cyclically. A'B'C' is the Dao triangle. (Reference: Preamble just before X(45738))

A-vertex coordinates:

Trilinears 0 : c*(a+b)*(a-b+c)^2 : b*(a+c)*(a+b-c)^2

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 A₁, B₁, C₁ of the circles {{X(6),B,C}}, {{X(6),C,A}}, {{X(6),A,B,}}, respectively, are the vertices of the 1st 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 A₂ = B_CB_A∩C_AC_B, and define B₂ and C₂ cyclically. The triangle A₂B₂C₂ is the 2nd Ehrmann triangle. (Reference: Preamble just before X8537)

A-vertex coordinates:

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

Ehrmann circumscribing

Let T_C1=A₁B₁C₁ be the triangle obtained by rotating ABC about the 1st Ehrmann pivot, P(5), by an angle of 2π/3, so that T_C1 circumscribes ABC. Triangle T_C1 is the 1st Ehrmann circumscribing triangle. Let T_C2=A₂B₂C₂ be the triangle obtained by rotating ABC about the 2nd Ehrmann pivot, U(5), by an angle of -2π/3, so that T_C2 circumscribes ABC. Triangle T_C2 is the 2nd Ehrmann circumscribing triangle. (Reference: Preamble just before X18300)

Note: Neither of both Ehrmann circumscribing triangles is a central triangle.

A-vertex coordinates:

1st Ehrmann circumscribing triangle:
Trilinears -(S^2-sqrt(3)*(SB-SC)*S-3*SB*SC)/a : (S+sqrt(3)*SA)*(S+sqrt(3)*SC)/b : (S-sqrt(3)*SA)*(S-sqrt(3)*SB)/c

2nd Ehrmann circumscribing triangle:
Trilinears -(S^2+sqrt(3)*(SB-SC)*S-3*SB*SC)/a : (S-sqrt(3)*SA)*(S-sqrt(3)*SC)/b : (S+sqrt(3)*SA)*(S+sqrt(3)*SB)/c

Ehrmann cross

The cross-triangle of the 1st and 2nd Ehrmann inscribed triangles is the Ehrman cross-triangle. (Reference: Preamble just before X18300)

Note: if T₁=A₁B₁C₁ and T₂=A₂B₂C₂ are two distinct triangles, the cross-triangle T*=A*B*C* of T₁ and T₂ is the triangle having vertices A*=B₁C₂ ∩ B₂C₁, B*=C₁A₂ ∩ C₂A₁ and C*=A₁B₂ ∩ A₂B₁.

Note: The Ehrmann cross triangle is degenerate, its vertices lying on line X(3)X(523) (the trilinear polar of X(2986)).

A-vertex coordinates:

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

Ehrmann inscribed

Let T_I1=A₁B₁C₁ be the triangle obtained by rotating ABC about the 1st Ehrmann pivot, P(5), by an angle of -2π/3, so that T_I1 is inscribed in ABC. Triangle T_I1 is the 1st Ehrmann inscribed triangle. Let T_I2=A₂B₂C₂ be the triangle obtained by rotating ABC about the 2nd Ehrmann pivot, U(5), by an angle of 2π/3, so that T_I2 is inscribed in ABC. Triangle T_I2 is the 2nd Ehrmann inscribed triangle. (Reference: Preamble just before X18300)

Note: Neither of both Ehrmann inscribed triangles is a central triangle.

A-vertex coordinates:

1st Ehrmann inscribed triangle:
Trilinears 0 : (S-sqrt(3)*SA)*c : (S+sqrt(3)*SA)*b

2nd Ehrmann inscribed triangle:
Trilinears 0 : (S+sqrt(3)*SA)*c : (S-sqrt(3)*SA)*b

Ehrmann mid

The mid-triangle of the 1st and 2nd Ehrmann circumscribing triangles is the Ehrman mid-triangle. (Reference: Preamble just before X18300)

Note: if T₁=A₁B₁C₁ and T₂=A₂B₂C₂ are two distinct triangles, the mid-triangle of T₁ and T₂ is the triangle having vertices the midpoints of {A₁,A₂}, {B₁,B₂} and {C₁,C₂}.

Equivalences:

(anti-X3-ABC reflections)-of-(Johnson), (Johnson)-of-(Euler).

A-vertex coordinates:

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

Ehrmann-side

The side-triangle of the 1st and 2nd Ehrmann circumscribing triangles is the Ehrmann-side triangle. (Reference: Preamble just before X18300)

Note: if T₁=A₁B₁C₁ and T₂=A₂B₂C₂ are two distinct triangles, the side-triangle T*=A*B*C* of T₁ and T₂ is the triangle having vertices A*=B₁B₂ ∩ C₁C₂, B*=C₁C₂ ∩ A₁A₂ and C*=A₁A₂ ∩ B₁B₂.

Equivalences:

(circumorthic)-of-(Johnson), (2nd Euler)-of-(X3-ABC reflections), (orthic)-of-(anti-Ehrmann-mid).

Only for acute ABC: (Aquila)-of-(2nd Euler).

A-vertex coordinates:

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

Ehrmann vertex

The vertex-triangle of the 1st and 2nd Ehrmann circumscribing triangles is the Ehrman vertex-triangle. (Reference: Preamble just before X18300)

Note: if T₁=A₁B₁C₁ and T₂=A₂B₂C₂ are two distinct triangles, the vertex-triangle T*=A*B*C* of T₁ and T₂ is the triangle having vertices A*=B₁C₁ ∩ B₂C₂, B*=C₁A₁ ∩ C₂A₂ and C*=A₁B₁ ∩ A₂B₂.

Equivalences:

(Kosnita)-of-(Johnson), (tangential)-of-(Ehrmann-mid).

A-vertex coordinates:

Trilinears -((4*SA+SB+SC)*S^2-(3*(SB+SC))*SA^2)/(a*(S^2-3*SA^2)) : SC/b : SB/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 triangle of ABC. (Reference: MathWorld -- Euler Triangle.)
A*, B*, C* are also the reflections of the vertices of the medial triangle in X(5).

Equivalences:

complement of (ABC-X3 reflections).

(ABC-X3 reflections)-of-(medial), (1st anti-circumperp)-of-(4th Euler), (circumorthic)-of-(3rd Euler), (2nd Euler)-of-(Wasat), (Johnson)-of-(Ehrmann-mid), (medial)-of-(Johnson).

Only for acute ABC: (1st circumperp)-of-(2nd Euler), (2nd circumperp)-of-(orthic), (4th Euler)-of-(anti-Wasat), (Hutson intouch)-of-(anti-Ursa minor).

A-vertex coordinates:

Trilinears 2((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)

Equivalences:

complement of (circumorthic).

(ABC-X3 reflections)-of-(orthic), (1st anti-circumperp)-of-(Euler), (6th anti-mixtilinear)-of-(X3-ABC reflections), (circumorthic)-of-(medial), (Ehrmann-side)-of-(anti-X3-ABC reflections), (Euler)-of-(anti-Wasat), (orthic)-of-(Johnson).

Only for acute ABC: (anti-Aquila)-of-(Ehrmann-side), (2nd anti-circumperp-tangential)-of-(anti-Ursa minor), (Aquila)-of-(6th anti-mixtilinear).

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)

Equivalences:

complement of (1st circumperp).

(ABC-X3 reflections)-of-(4th Euler), (1st circumperp)-of-(medial), (2nd circumperp)-of-(Euler), (4th Euler)-of-(Johnson), (medial)-of-(Wasat).

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)

Equivalences:

complement of (2nd circumperp).

(ABC-X3 reflections)-of-(3rd Euler), (1st circumperp)-of-(Euler), (2nd circumperp)-of-(medial), (Euler)-of-(Wasat), (3rd Euler)-of-(Johnson).

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)

Equivalences:

complement of (circummedial).

A-vertex coordinates:

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

excenters-incenter reflections

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

Note: The original name of this triangle is T(-1,3). Reference: TCCT, p. 173.

Equivalences:

(Ara)-of-(Hutson intouch), (excentral)-of-(5th mixtilinear).

A-vertex coordinates:

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

excenters-midpoints

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

Note: The original name of this triangle is T(-2,1). Reference: TCCT, p. 173.

Equivalences:

complement of (Garcia-reflection).

A-vertex coordinates:

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

excenters-reflections

See excenters-incenter reflections.

excentral

The triangle whose vertices are the excenters of ABC.

Equivalences:

anticomplement of (Wasat), complement of (2nd Conway).

(anticomplementary)-of-(1st circumperp), (anti-Euler)-of-(2nd circumperp), (Ara)-of-(intouch), (2nd circumperp)-of-(Aquila), (2nd Conway)-of-(medial), (hexyl)-of-(ABC-X3 reflections), (Johnson)-of-(hexyl), (medial)-of-(6th mixtilinear), (Ursa-minor)-of-(anti-Mandart-incircle), (Wasat)-of-(anticomplementary).

Note: The excentral triangle is the anticevian and the antipedal triangle of X(1).

A-vertex coordinates:

Trilinears -1 : 1 : 1

1st and 2nd excosine

Let A_b, A_c be the points where the A-excosine circle cuts again the sidelines AC and AB, respectively, and build {B_c, B_a}, {C_a, C_b} cyclically. The triangle A₁B₁C₁ bounded by the lines A_bA_c, B_cB_a and C_aC_b is the 1st excosine triangle of ABC and the triangle A₂B₂C₂ bounded by the lines B_cC_b, C_aA_c and A_bB_a is the 2nd excosine triangle of ABC. (Reference: X17807)

Note: If the tangents at B and C to the circumcircle of a triangle ABC intersect at A', then the circle with center A' passing through B and C is called A-excosine circle of ABC. This circle cuts AB and AC again at two points which are the extremities of a diameter of it. (Reference: MathWorld -- Excosine Circle.)

Equivalences:

1st excosine = (anticomplementary)-of-(tangential), (anti-Wasat)-of-(Ara).

A-vertex coordinates:

1st excosine triangle:
Trilinears -S^2*a : (S^2-2*SA*SB)*b : (S^2-2*SA*SC)*c

2nd excosine triangle:
Trilinears -S^2*SB*SC/(SA*a) : (S^2-2*SA*SB)*b : (S^2-2*SA*SC)*c

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 triangle of ABC.

Equivalences:

anticomplement of (1st Zaniah), complement of (Gemini 29).

(Aries)-of-(intouch).

A-vertex coordinates:

Trilinears 0 : 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 A₂ = B_CB_A∩C_AC_B, and define B₂ and C₂ cyclically. The 2nd extouch triangle is A₂B₂C₂. (Reference: X5927)

Equivalences:

anticomplement of (Ascella), complement of (Conway).

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 A₃ = C_AA_C∩A_BB_A, and define B₃ and C₃ cyclically. The 3rd extouch triangle is A₃B₃C₃. (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 A₄ = B_CC_A∩B_AC_B, and define B₄ and C₄ cyclically. The 4th extouch triangle is A₄B₄C₄. (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 D₁ = B_BB_C∩C_CC_A, and define D₂ and D₃ cyclically. Let E₁ = B_BB_A∩C_CC_B, and define E₂ and E₃ cyclically. Define A₅ = D₂E₂∩D₃E₃, and define B₅ and C₅ cyclically. The 5th extouch triangle is A₅B₅C₅. (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:

inner-Fermat

In a triangle ABC build equilateral BCA', CAB', ABC' inwards ABC. Then A'B'C' is the inner-Fermat triangle of ABC.

Equivalences:

anticomplement of (1st half-diamonds).

(anticomplementary)-of-(1st half-diamonds).

A-vertex coordinates:

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

outer-Fermat

In a triangle ABC build equilateral BCA', CAB', ABC' outwards ABC. Then A'B'C' is the outer-Fermat triangle of ABC.

Equivalences:

anticomplement of (2nd half-diamonds).

(anticomplementary)-of-(2nd half-diamonds).

A-vertex coordinates:

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

1st and 2nd Fermat-Dao equilateral

Let F=X(13) be the 1st Fermat point of ABC and L_a the parallel to BC through F. Let A_b and A_c be the points, others than F, where L_a cuts the circumcircles of FAC and FAB, respectively. Construct B_c, B_a and C_a, C_b cyclically. Let A'=A_bC_b∩A_cB_c and similarly B' and C'. The triangle A'B'C' is equilateral and is named the 1st Fermat-Dao equilateral triangle. (Reference: X16247)

If the above construction is done from F=X(14) (the 2nd Fermat point of ABC), the obtained triangle A'B'C' is also equilateral and is called the 2nd Fermat-Dao equilateral triangle. (Reference: X16247)

You can see a sketch of 1st and 2nd Fermat-Dao equilateral triangles here.

A-vertex coordinates:

1st Fermat-Dao equilateral triangle:
Trilinears
-a*(sqrt(3)*S*((-6*R^2+3*SA+12*SW)*S^2+(7*SA^2+SB*SC+SW^2)*SW+18*R^2*SB*SC)+(20*S^2-6*R^2*(3*SA-SW)+19*SA^2-SB*SC+11*SW^2)*S^2+3*(2*SA-SW)*SA*SW^2)/(sqrt(3)*a^2+2*S) : (S+sqrt(3)*SB)*(S^2+sqrt(3)*(6*R^2-SW+SC)*S+SW*SC)*b : (S+sqrt(3)*SC)*(S^2+sqrt(3)*(6*R^2-SW+SB)*S+SB*SW)*c

2nd Fermat-Dao equilateral triangle:
Trilinears
-a*(-sqrt(3)*S*((-6*R^2+3*SA+12*SW)*S^2+(7*SA^2+SB*SC+SW^2)*SW+18*R^2*SB*SC)+(20*S^2-6*R^2*(3*SA-SW)+19*SA^2-SB*SC+11*SW^2)*S^2+3*(2*SA-SW)*SA*SW^2)/(sqrt(3)*a^2-2*S) : (-S+sqrt(3)*SB)*(S^2-sqrt(3)*(6*R^2-SW+SC)*S+SW*SC)*b : (-S+sqrt(3)*SC)*(S^2-sqrt(3)*(6*R^2-SW+SB)*S+SB*SW)*c

3rd and 4th Fermat-Dao equilateral

Let F=X(13) be the 1st Fermat point of ABC. Let A_b and A_c be points on AC and AB, respectively, such that FA_bA_c is an equilateral triangle with the same orientation as ABC. Denote the centroid of FA_bA_c as A' and similarly build B' and C'. The triangle A'B'C' is equilateral and is named the 3rd Fermat-Dao equilateral triangle. (Reference: X16267)

Let F=X(14) be the 2nd Fermat point of ABC. Let A_b and A_c be points on AC and AB, respectively, such that FA_bA_c is an equilateral triangle with opposite orientation than ABC. Denote the centroid of FA_bA_c as A" and similarly build B" and C". The triangle A"B"C" is equilateral and is named the 4th Fermat-Dao equilateral triangle. (Reference: X16268)

You can see a sketch of 3rd and 4th Fermat-Dao equilateral triangles here.

A-vertex coordinates:

3rd Fermat-Dao equilateral triangle:
Trilinears
(sqrt(3)*(3*S^2+SB*SC)+5*(SB+SC)*S)/(a*(sqrt(3)*SA+S)) : (SC+sqrt(3)*S)/b : (SB+sqrt(3)*S)/c

4th Fermat-Dao equilateral triangle:
Trilinears
(sqrt(3)*(3*S^2+SB*SC)-5*(SB+SC)*S)/(a*(sqrt(3)*SA-S)) : (SC-sqrt(3)*S)/b : (SB-sqrt(3)*S)/c

5th and 6th Fermat-Dao equilateral

Let F=X(13) be the 1st Fermat point of ABC. Let B_a and C_a be points on BC such that the triangle FB_aC_a is equilateral. Define the pairs {C_b, A_b} and {A_b, B_c}cyclically. Let A' be the point other than F that lies on the circles (FA_cB_c) and (FC_bA_b), and define B' and C' cyclically. Then the triangle A'B'C' is equilateral and is the 5th Fermat Dao equilateral triangle of ABC. (Reference: X16459)

When F=X(14), i.e, the 2nd Fermat point of ABC, the obtained triangle A'B'C' is also equilateral and is named the 6th Fermat-Dao equilateral triangle. (Reference: X16459)

You can see a sketch of 5th and 6th Fermat-Dao equilateral triangles here.

A-vertex coordinates:

5th Fermat-Dao equilateral triangle:
Trilinears
((7*SA+4*SW)*S^2+3*SA^3+5*sqrt(3)*(S^2+SA^2)*S)/(a*(sqrt(3)*SA+S)^2) : b*(sqrt(3)*c^2+2*S)/(sqrt(3)*SB+S) : c*(sqrt(3)*b^2+2*S)/(sqrt(3)*SC+S)

6th Fermat-Dao equilateral triangle:
Trilinears
((7*SA+4*SW)*S^2+3*SA^3-5*sqrt(3)*(S^2+SA^2)*S)/(a*(sqrt(3)*SA-S)^2) : b*(sqrt(3)*c^2-2*S)/(sqrt(3)*SB-S) : c*(sqrt(3)*b^2-2*S)/(sqrt(3)*SC-S)

7th and 8th Fermat-Dao equilateral

Let F=X(13) be the 1st Fermat point of ABC. Let A_b and A_c be points on AC and AB, respectively, such that FA_bA_c is an equilateral triangle with the same orientation as ABC. Let A' be the midpoint of A_b and A_c and define B' and C' cyclically. The triangle A'B'C' is equilateral and is named the 7th Fermat-Dao equilateral triangle. (Reference: X396)

Let F=X(14) be the 2nd Fermat point of ABC. Let A_b and A_c be points on AC and AB, respectively, such that FA_bA_c is an equilateral triangle with opposite orientation than ABC. Let A" be the midpoint of A_b and A_c and define B" and C" cyclically. The triangle A"B"C" is equilateral and is named the 8th Fermat-Dao equilateral triangle. (Reference: X395)

You can see a sketch of 7th and 8th Fermat-Dao equilateral triangles here.

A-vertex coordinates:

7th Fermat-Dao equilateral triangle:
Trilinears
2*(sqrt(3)*(SB+SC)+2*S)/(a*(sqrt(3)*SA+S)) : 1/b : 1/c

8th Fermat-Dao equilateral triangle:
Trilinears
2*(sqrt(3)*(SB+SC)-2*S)/(a*(sqrt(3)*SA-S)) : 1/b : 1/c

9th and 10th Fermat-Dao equilateral

Let F = X(13) be the 1st Fermat point of ABC. Let B_a, C_a be the points on BC such that FB_aC_a is an equilateral triangle. Define C_b, A_b and A_c, B_c cyclically. Let A', B', C' be the midpoints of A_bA_c, B_cB_a, C_aC_b, respectively. The triangle A'B'C' is equilateral and is named here the 9th Dao-Fermat equilateral triangle. (Reference: X16536)

When F=X(14), i.e., the 2nd Fermat point of ABC, the obtained triangle A'B'C' is also equilateral and is called the 10th Fermat-Dao equilateral triangle. (Reference: X16536)

You can see a sketch of 9th and 10th Fermat-Dao equilateral triangles here.

A-vertex coordinates:

9th Fermat-Dao equilateral triangle:
Trilinears
2*S*((2*S^2+SA^2+SB*SC)*sqrt(3)+(6*R^2+SA+SW)*S)/(a*(sqrt(3)*SA+S)^2) : b : c

10th Fermat-Dao equilateral triangle:
Trilinears
-2*S*((2*S^2+SA^2+SB*SC)*sqrt(3)-(6*R^2+SA+SW)*S)/(a*(sqrt(3)*SA-S)^2) : b : c

11th and 12th Fermat-Dao equilateral

Let F = X(13) be the 1st Fermat point of ABC. Let A_b and A_c be the points where the circle FBC cuts again AC and AB, respectively. Define B_c, B_a and C_a, C_b cyclically. Denote A', B', C' the circumcenters of AB_cC_b, BC_aA_c and CB_aA_b, respectively. Then the triangle A'B'C' is equilateral and is here named the 11th Dao-Fermat equilateral triangle.

When F=X(14), i.e., the 2nd Fermat point of ABC, the resulting triangle is also equilateral and is here named the 12th Fermat-Dao equilateral triangle. (Reference: X16626)

A-vertex coordinates:

11th Fermat-Dao equilateral triangle:
Trilinears
(sqrt(3)*(S^2+SB*SC)+(SB+SC)*S)/(a*(sqrt(3)*SA-S)) : SC/b : SB/c

12th Fermat-Dao equilateral triangle:
Trilinears
(sqrt(3)*(S^2+SB*SC)-(SB+SC)*S)/(a*(sqrt(3)*SA+S)) : SC/b : SB/c

13th and 14th Fermat-Dao equilateral

Let F = X(13) be the 1st Fermat point of ABC. Let A_b and A_c be the points where the circle FBC cuts again AC and AB, respectively. Define B_c, B_a and C_a, C_b cyclically. Let A', B', C' be centroids of AB_cC_b, BC_aA_c, CA_bB_a respectively. Then the triangle A'B'C' is equilateral and is the 13th Fermat-Dao equilateral triangle.

When F=X(14), i.e., the 2nd Fermat point of ABC, the resulting triangle is also equilateral and is the 14th Fermat-Dao equilateral triangle. (Reference: X16636)

A-vertex coordinates:

13th Fermat-Dao equilateral triangle:
Trilinears
(sqrt(3)*(2*S^2+SA^2+SB*SC)+(SA+SW)*S)/(a*(sqrt(3)*SA-S)) : b : c

14th Fermat-Dao equilateral triangle:
Trilinears
(sqrt(3)*(2*S^2+SA^2+SB*SC)-(SA+SW)*S)/(a*(sqrt(3)*SA+S)) : b : c

15th and 16th Fermat-Dao equilateral

Let A_oB_oC_o be the outer-Fermat triangle of ABC. Denote p'_a the polar of A with respect to the circle with center A_o and radius a=BC. Define similarly p'_b and p'_c. Then the triangle A'B'C' bounded by these polars is equilateral and is the 15th Fermat-Dao equilateral triangle.

Let A_iB_iC_i be the inner-Fermat triangle of ABC. Denote p"_a the polar of A with respect to the circle with center A_i and radius a=BC. Define similarly p"_b and p"_c. Then the triangle A"B"C" bounded by these polars is equilateral and is the 16th Fermat-Dao equilateral triangle. (Reference: X16960)

Equivalences:

15th Fermat-Dao = (anticomplementary)-of-(1st isodynamic-Dao), (anti-Ehrmann-mid)-of-(1st isodynamic-Dao), (3rd anti-Euler)-of-(1st isodynamic-Dao), (anti-inverse-in-incircle)-of-(3rd isodynamic-Dao), (anti-Mandart-incircle)-of-(1st isodynamic-Dao), (Ara)-of-(3rd isodynamic-Dao), (2nd Conway)-of-(3rd isodynamic-Dao), (excentral)-of-(1st isodynamic-Dao), (1st excosine)-of-(3rd isodynamic-Dao), (6th mixtilinear)-of-(3rd isodynamic-Dao), (reflection)-of-(1st isodynamic-Dao), (tangential)-of-(1st isodynamic-Dao).

16th Fermat-Dao = (anticomplementary)-of-(2nd isodynamic-Dao), (anti-Ehrmann-mid)-of-(2nd isodynamic-Dao), (3rd anti-Euler)-of-(2nd isodynamic-Dao), (anti-inverse-in-incircle)-of-(4th isodynamic-Dao), (anti-Mandart-incircle)-of-(2nd isodynamic-Dao), (Ara)-of-(4th isodynamic-Dao), (2nd Conway)-of-(4th isodynamic-Dao), (excentral)-of-(2nd isodynamic-Dao), (1st excosine)-of-(4th isodynamic-Dao), (6th mixtilinear)-of-(4th isodynamic-Dao), (reflection)-of-(2nd isodynamic-Dao), (tangential)-of-(2nd isodynamic-Dao).

A-vertex coordinates:

15th Fermat-Dao equilateral triangle:
Trilinears
(sqrt(3)*(3*S^2+SB*SC)+5*(SB+SC)*S)/(a*(SA-sqrt(3)*S)) : (sqrt(3)*SC+S)/b : (sqrt(3)*SB+S)/c

16th Fermat-Dao equilateral triangle:
Trilinears
(sqrt(3)*(3*S^2+SB*SC)-5*(SB+SC)*S)/(a*(SA+sqrt(3)*S)) : (sqrt(3)*SC-S)/b : (sqrt(3)*SB-S)/c

1st to 4th inner- and outer- Fermat-Dao-Nhi

Let ABC be a triangle with centroid G and inner-Fermat (or outer-Fermat) triangle A_fB_fC_f.

Let A* be the reflection of G in A_f and A₁ the reflection of A* in A; build B₁ and C₁ cyclically.

Let A* be the reflection of G in A and A₂ the reflection of A* in A_f; build B₂ and C₂ cyclically.

Let A* be the reflection of A in G and A₃ the reflection of A* in A_f; build B₃ and C₃ cyclically.

Let A* be the reflection of A_f in G and A₄ the reflection of A* in A; build B₄ and C₄ cyclically.

Then triangles A₁B₁C₁, A₂B₂C₂, A₃B₃C₃, A₄B₄C₄ are equilateral and perspective to ABC. (Reference: Preamble just before X33602.)
These triangles built from the inner-Fermat triangle are the 1st, 2nd, 3rd and 4th inner-Fermat-Dao-Nhi triangles, respectively, and their correspondents built from the outer-Fermat triangle are the 1st, 2nd, 3rd and 4th outer-Fermat-Dao-Nhi triangles, respectively.

Equivalences:

1st inner-Fermat-Dao-Nhi = anticomplement of (3rd outer-Fermat-Dao-Nhi).

1st inner-Fermat-Dao-Nhi = (ABC-X3 reflections)-of-(2nd inner-Fermat-Dao-Nhi), (anticomplementary)-of-(3rd outer-Fermat-Dao-Nhi), (anti-Ehrmann-mid)-of-(3rd outer-Fermat-Dao-Nhi), (anti-Euler)-of-(4th outer-Fermat-Dao-Nhi), (3rd anti-Euler)-of-(3rd outer-Fermat-Dao-Nhi), (4th anti-Euler)-of-(4th outer-Fermat-Dao-Nhi), (anti-Hutson intouch)-of-(4th outer-Fermat-Dao-Nhi), (anti-Mandart-incircle)-of-(3rd outer-Fermat-Dao-Nhi), (Aquila)-of-(4th outer-Fermat-Dao-Nhi), (circumorthic)-of-(2nd inner-Fermat-Dao-Nhi), (2nd circumperp tangential)-of-(4th outer-Fermat-Dao-Nhi), (excentral)-of-(3rd outer-Fermat-Dao-Nhi), (outer-Garcia)-of-(2nd inner-Fermat-Dao-Nhi), (hexyl)-of-(4th outer-Fermat-Dao-Nhi), (Johnson)-of-(2nd inner-Fermat-Dao-Nhi), (Kosnita)-of-(2nd inner-Fermat-Dao-Nhi), (reflection)-of-(3rd outer-Fermat-Dao-Nhi), (tangential)-of-(3rd outer-Fermat-Dao-Nhi), (X3-ABC reflections)-of-(4th outer-Fermat-Dao-Nhi).

2nd inner-Fermat-Dao-Nhi = anticomplement of (4th outer-Fermat-Dao-Nhi).

2nd inner-Fermat-Dao-Nhi = (ABC-X3 reflections)-of-(1st inner-Fermat-Dao-Nhi), (anticomplementary)-of-(4th outer-Fermat-Dao-Nhi), (anti-Ehrmann-mid)-of-(4th outer-Fermat-Dao-Nhi), (anti-Euler)-of-(3rd outer-Fermat-Dao-Nhi), (3rd anti-Euler)-of-(4th outer-Fermat-Dao-Nhi), (4th anti-Euler)-of-(3rd outer-Fermat-Dao-Nhi), (anti-Hutson intouch)-of-(3rd outer-Fermat-Dao-Nhi), (anti-Mandart-incircle)-of-(4th outer-Fermat-Dao-Nhi), (Aquila)-of-(3rd outer-Fermat-Dao-Nhi), (circumorthic)-of-(1st inner-Fermat-Dao-Nhi), (2nd circumperp tangential)-of-(3rd outer-Fermat-Dao-Nhi), (excentral)-of-(4th outer-Fermat-Dao-Nhi), (outer-Garcia)-of-(1st inner-Fermat-Dao-Nhi), (hexyl)-of-(3rd outer-Fermat-Dao-Nhi), (Johnson)-of-(1st inner-Fermat-Dao-Nhi), (Kosnita)-of-(1st inner-Fermat-Dao-Nhi), (reflection)-of-(4th outer-Fermat-Dao-Nhi), (tangential)-of-(4th outer-Fermat-Dao-Nhi), (X3-ABC reflections)-of-(3rd outer-Fermat-Dao-Nhi).

3rd inner-Fermat-Dao-Nhi = complement of (1st outer-Fermat-Dao-Nhi).

3rd inner-Fermat-Dao-Nhi = (ABC-X3 reflections)-of-(4th inner-Fermat-Dao-Nhi), (anti-Aquila)-of-(2nd outer-Fermat-Dao-Nhi), (2nd anti-circumperp-tangential)-of-(2nd outer-Fermat-Dao-Nhi), (anti-X3-ABC reflections)-of-(2nd outer-Fermat-Dao-Nhi), (circumorthic)-of-(4th inner-Fermat-Dao-Nhi), (Ehrmann-mid)-of-(1st outer-Fermat-Dao-Nhi), (Euler)-of-(2nd outer-Fermat-Dao-Nhi), (2nd Euler)-of-(2nd outer-Fermat-Dao-Nhi), (3rd Euler)-of-(1st outer-Fermat-Dao-Nhi), (4th Euler)-of-(2nd outer-Fermat-Dao-Nhi), (outer-Garcia)-of-(4th inner-Fermat-Dao-Nhi), (Hutson intouch)-of-(2nd outer-Fermat-Dao-Nhi), (Johnson)-of-(4th inner-Fermat-Dao-Nhi), (Kosnita)-of-(4th inner-Fermat-Dao-Nhi), (Mandart-incircle)-of-(1st outer-Fermat-Dao-Nhi).

4th inner-Fermat-Dao-Nhi = complement of (2nd outer-Fermat-Dao-Nhi).

4th inner-Fermat-Dao-Nhi = (ABC-X3 reflections)-of-(3rd inner-Fermat-Dao-Nhi), (anti-Aquila)-of-(1st outer-Fermat-Dao-Nhi), (2nd anti-circumperp-tangential)-of-(1st outer-Fermat-Dao-Nhi), (anti-X3-ABC reflections)-of-(1st outer-Fermat-Dao-Nhi), (circumorthic)-of-(3rd inner-Fermat-Dao-Nhi), (Ehrmann-mid)-of-(2nd outer-Fermat-Dao-Nhi), (Euler)-of-(1st outer-Fermat-Dao-Nhi), (2nd Euler)-of-(1st outer-Fermat-Dao-Nhi), (3rd Euler)-of-(2nd outer-Fermat-Dao-Nhi), (4th Euler)-of-(1st outer-Fermat-Dao-Nhi), (outer-Garcia)-of-(3rd inner-Fermat-Dao-Nhi), (Hutson intouch)-of-(1st outer-Fermat-Dao-Nhi), (Johnson)-of-(3rd inner-Fermat-Dao-Nhi), (Kosnita)-of-(3rd inner-Fermat-Dao-Nhi), (Mandart-incircle)-of-(2nd outer-Fermat-Dao-Nhi).

1st outer-Fermat-Dao-Nhi = anticomplement of (3rd inner-Fermat-Dao-Nhi).

1st outer-Fermat-Dao-Nhi = (ABC-X3 reflections)-of-(2nd outer-Fermat-Dao-Nhi), (anticomplementary)-of-(3rd inner-Fermat-Dao-Nhi), (anti-Ehrmann-mid)-of-(3rd inner-Fermat-Dao-Nhi), (anti-Euler)-of-(4th inner-Fermat-Dao-Nhi), (3rd anti-Euler)-of-(3rd inner-Fermat-Dao-Nhi), (4th anti-Euler)-of-(4th inner-Fermat-Dao-Nhi), (anti-Hutson intouch)-of-(4th inner-Fermat-Dao-Nhi), (anti-Mandart-incircle)-of-(3rd inner-Fermat-Dao-Nhi), (Aquila)-of-(4th inner-Fermat-Dao-Nhi), (circumorthic)-of-(2nd outer-Fermat-Dao-Nhi), (2nd circumperp tangential)-of-(4th inner-Fermat-Dao-Nhi), (excentral)-of-(3rd inner-Fermat-Dao-Nhi), (outer-Garcia)-of-(2nd outer-Fermat-Dao-Nhi), (hexyl)-of-(4th inner-Fermat-Dao-Nhi), (Johnson)-of-(2nd outer-Fermat-Dao-Nhi), (Kosnita)-of-(2nd outer-Fermat-Dao-Nhi), (reflection)-of-(3rd inner-Fermat-Dao-Nhi), (tangential)-of-(3rd inner-Fermat-Dao-Nhi), (X3-ABC reflections)-of-(4th inner-Fermat-Dao-Nhi).

2nd outer-Fermat-Dao-Nhi = anticomplement of (4th inner-Fermat-Dao-Nhi).

2nd outer-Fermat-Dao-Nhi = (ABC-X3 reflections)-of-(1st outer-Fermat-Dao-Nhi), (anticomplementary)-of-(4th inner-Fermat-Dao-Nhi), (anti-Ehrmann-mid)-of-(4th inner-Fermat-Dao-Nhi), (anti-Euler)-of-(3rd inner-Fermat-Dao-Nhi), (3rd anti-Euler)-of-(4th inner-Fermat-Dao-Nhi), (4th anti-Euler)-of-(3rd inner-Fermat-Dao-Nhi), (anti-Hutson intouch)-of-(3rd inner-Fermat-Dao-Nhi), (anti-Mandart-incircle)-of-(4th inner-Fermat-Dao-Nhi), (Aquila)-of-(3rd inner-Fermat-Dao-Nhi), (circumorthic)-of-(1st outer-Fermat-Dao-Nhi), (2nd circumperp tangential)-of-(3rd inner-Fermat-Dao-Nhi), (excentral)-of-(4th inner-Fermat-Dao-Nhi), (outer-Garcia)-of-(1st outer-Fermat-Dao-Nhi), (hexyl)-of-(3rd inner-Fermat-Dao-Nhi), (Johnson)-of-(1st outer-Fermat-Dao-Nhi), (Kosnita)-of-(1st outer-Fermat-Dao-Nhi), (reflection)-of-(4th inner-Fermat-Dao-Nhi), (tangential)-of-(4th inner-Fermat-Dao-Nhi), (X3-ABC reflections)-of-(3rd inner-Fermat-Dao-Nhi).
3rd outer-Fermat-Dao-Nhi = complement of (1st inner-Fermat-Dao-Nhi).

3rd outer-Fermat-Dao-Nhi = (ABC-X3 reflections)-of-(4th outer-Fermat-Dao-Nhi), (anti-Aquila)-of-(2nd inner-Fermat-Dao-Nhi), (2nd anti-circumperp-tangential)-of-(2nd inner-Fermat-Dao-Nhi), (anti-X3-ABC reflections)-of-(2nd inner-Fermat-Dao-Nhi), (circumorthic)-of-(4th outer-Fermat-Dao-Nhi), (Ehrmann-mid)-of-(1st inner-Fermat-Dao-Nhi), (Euler)-of-(2nd inner-Fermat-Dao-Nhi), (2nd Euler)-of-(2nd inner-Fermat-Dao-Nhi), (3rd Euler)-of-(1st inner-Fermat-Dao-Nhi), (4th Euler)-of-(2nd inner-Fermat-Dao-Nhi), (outer-Garcia)-of-(4th outer-Fermat-Dao-Nhi), (Hutson intouch)-of-(2nd inner-Fermat-Dao-Nhi), (Johnson)-of-(4th outer-Fermat-Dao-Nhi), (Kosnita)-of-(4th outer-Fermat-Dao-Nhi), (Mandart-incircle)-of-(1st inner-Fermat-Dao-Nhi).

4th outer-Fermat-Dao-Nhi = complement of (2nd inner-Fermat-Dao-Nhi).

4th outer-Fermat-Dao-Nhi = (ABC-X3 reflections)-of-(3rd outer-Fermat-Dao-Nhi), (anti-Aquila)-of-(1st inner-Fermat-Dao-Nhi), (2nd anti-circumperp-tangential)-of-(1st inner-Fermat-Dao-Nhi), (anti-X3-ABC reflections)-of-(1st inner-Fermat-Dao-Nhi), (circumorthic)-of-(3rd outer-Fermat-Dao-Nhi), (Ehrmann-mid)-of-(2nd inner-Fermat-Dao-Nhi), (Euler)-of-(1st inner-Fermat-Dao-Nhi), (2nd Euler)-of-(1st inner-Fermat-Dao-Nhi), (3rd Euler)-of-(2nd inner-Fermat-Dao-Nhi), (4th Euler)-of-(1st inner-Fermat-Dao-Nhi), (outer-Garcia)-of-(3rd outer-Fermat-Dao-Nhi), (Hutson intouch)-of-(1st inner-Fermat-Dao-Nhi), (Johnson)-of-(3rd outer-Fermat-Dao-Nhi), (Kosnita)-of-(3rd outer-Fermat-Dao-Nhi), (Mandart-incircle)-of-(2nd inner-Fermat-Dao-Nhi).

A-vertices coordinates:

1th to 4th inner-Fermat-Dao-Nhi triangles
Trilinears A₁ = (7*S-3*sqrt(3)*a^2)/a : (-2*S+3*sqrt(3)*SC)/b : (-2*S+3*sqrt(3)*SB)/c
Trilinears A₂ = (5*S-3*sqrt(3)*a^2)/a : (-4*S+3*sqrt(3)*SC)/b : (-4*S+3*sqrt(3)*SB)/c
Trilinears A₃ = (-4*S-3*sqrt(3)*a^2)/a : ( 5*S+3*sqrt(3)*SC)/b : ( 5*S+3*sqrt(3)*SB)/c
Trilinears A₄ = (-8*S-3*sqrt(3)*a^2)/a : ( S+3*sqrt(3)*SC)/b : ( S+3*sqrt(3)*SB)/c

1th to 4th outer-Fermat-Dao-Nhi triangles
Trilinears A₁ = (-7*S-3*sqrt(3)*a^2)/a : ( 2*S+3*sqrt(3)*SC)/b : ( 2*S+3*sqrt(3)*SB)/c
Trilinears A₂ = (-5*S-3*sqrt(3)*a^2)/a : ( 4*S+3*sqrt(3)*SC)/b : ( 4*S+3*sqrt(3)*SB)/c
Trilinears A₃ = ( 4*S-3*sqrt(3)*a^2)/a : (-5*S+3*sqrt(3)*SC)/b : (-5*S+3*sqrt(3)*SB)/c
Trilinears A₄ = ( 8*S-3*sqrt(3)*a^2)/a : ( -S+3*sqrt(3)*SC)/b : ( -S+3*sqrt(3)*SB)/c

Feuerbach

The Feuerbach triangle is the triangle formed by the three points of tangency of the nine-point circle with the excircles. (Reference: MathWorld -- Feuerbach Triangle.)

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 triangle has vertices the reflections of the vertices of 2nd circumperp triangle on the sidelines of ABC.

A-vertex coordinates:

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

2nd Fuhrmann

The 2nd Fuhrmann triangle has vertices the reflections of the vertices of 1st circumperp triangle on the sidelines of ABC.

A-vertex coordinates:

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

G:

inner- and outer- Gallatly-Kiepert

inner-Gallatly-Kiepert: See 1st Brocard

outer-Gallatly-Kiepert: See 1st Brocard reflected

inner-Garcia

Let T_AT_BT_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 the inner Garcia triangle. (Reference: X5587)

A-vertex coordinates:

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

outer-Garcia

Let T_AT_BT_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 the inner Garcia triangle. (Reference: X5587)

Equivalences:

anticomplement of (anti-Aquila).

(anti-Aquila)-of-(anticomplementary), (Kosnita)-of-(inner-Conway), (medial)-of-(Aquila).

A-vertex coordinates:

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

Garcia-Moses

Let A'B'C' be the cevian triangle of X(1), M_a the midpoint of AA', and define M_b and M_c cyclically. Let A" = BM_c ∩ CM_b, and define B" and C" cyclicaly. The triangle A"B"C" is the García-Moses triangle. Reference: X51504)

A-vertex coordinates:

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

Garcia-reflection

The Garcia-reflection triangle has vertices the reflections of the excenters on the midpoints of their corresponding sides. (Reference: X15995)

Equivalences:

anticomplement of (excenters-midpoints).

A-vertex coordinates:

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

Gemini triangles 1 to 111

See here.

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_aC_aC, CC_bA_bA, AA_cB_cB on the sides BC, CA, AB of triangle ABC (inwards). Then, the triangle A'B'C' enclosed by the lines B_aC_a, C_bA_b, A_cB_c is the inner-Grebe triangle of ABC.

A-vertex coordinates:

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

outer-Grebe

Erect squares BB_aC_aC, CC_bA_bA, AA_cB_cB on the sides BC, CA, AB of triangle ABC (outwards). Then, the triangle A'B'C' enclosed by the lines B_aC_a, C_bA_b, A_cB_c is the outer-Grebe triangle of ABC. (Reference: Hyacinthos #6445)

A-vertex coordinates:

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

H:

half-altitude

See midheight triangle.

1st and 2nd half-diamonds

Build equilateral triangles AB'C', BC'A' and CA'B'. There are two triads of such triangles leading to two distinct triangles A'B'C', called the 1st and 2nd half-diamonds triangles of ABC. (Reference: Preamble just before of X33338.)

A construction: Let P=X(17) or P=X(18) (Napoleon points). Let B_c the reflection of B in the C-cevian of P and C_b the reflection of C in the B-cevian-of P. Denote as a' the circle {{A,B_c,C_b}} and build b', c', cyclically, naming their centers as A', B', C', respectively. Then, for P=X(18), A'B'C' is the 1st half-diamonds triangle, and, for P=X(17), A'B'C' is the 2nd half-diamonds triangle.

Equivalences:

1st half-diamonds triangle = complement of (inner-Fermat).

1st half-diamonds triangle = (medial)-of-(inner-Fermat).

2nd half-diamonds triangle = complement of (outer-Fermat).

2nd half-diamonds triangle = (medial)-of-(outer-Fermat).

A-vertex coordinates:

1st half-diamonds triangle:
Trilinears -(sqrt(3)*a^2-2*S)/a : (sqrt(3)*SC+S)/b : (sqrt(3)*SB+S)/c

2nd half-diamonds triangle:
Trilinears -(sqrt(3)*a^2+2*S)/a : (sqrt(3)*SC-S)/b : (sqrt(3)*SB-S)/c

1st and 2nd half-diamonds-central

The centers of the triangles AB'C', BC'A' and CA'B' in the construction of the 1st and 2nd half-diamonds triangles are vertices of equilateral triangles, here named the 1st and 2nd half-diamonds-central triangles of ABC. (Reference: Preamble just before of X33338.)

Equivalences:

1st half-diamonds-central = complement of (outer-Napoleon).

1st half-diamonds-central = (Ehrmann-mid)-of-(outer-Napoleon), (3rd Euler)-of-(outer-Napoleon), (Mandart-incircle)-of-(outer-Napoleon), (medial)-of-(outer-Napoleon).

2nd half-diamonds-central = complement of (inner-Napoleon).

A-vertex coordinates:

1st half-diamonds-central equilateral triangle:
Trilinears -(a^2+2*sqrt(3)*S)/a : (SC-sqrt(3)*S)/b : (SB-sqrt(3)*S)/c

2nd half-diamonds-central equilateral triangle:
Trilinears -(a^2-2*sqrt(3)*S)/a : (SC+sqrt(3)*S)/b : (SB+sqrt(3)*S)/c

1st and 2nd half-squares

Build isosceles rectangle triangles AB'C', BC'A' and CA'B', having vertical angles at A, B, C, respectively. There are two triads of such triangles leading to two distinct triangles A'B'C', called the 1st and 2nd half-squares triangles of ABC. (Reference: Preamble just before of X33338.)

Equivalences:

1st half-squares = anticomplement of (outer-Vecten).

1st half-squares = (anticomplementary)-of-(outer-Vecten), (6th mixtilinear)-of-(3rd outer-Vecten).

2nd half-squares = anticomplement of (inner-Vecten).

2nd half-squares = (anticomplementary)-of-(inner-Vecten).

A-vertex coordinates:

1st half-squares triangle:
Trilinears -(a^2+S)/a : SC/b : SB/c

2nd half-squares triangle:
Trilinears -(a^2-S)/a : SC/b : SB/c

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 A₀ be the point of intersection of lines BC_A and CB_A. Define B₀ and C₀ cyclically. A₀B₀C₀ is the 1st 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 A₀B₀C₀ be the 1st Hatzipolakis triangle of ABC. Let A₁ be the orthogonal projection of A₀ onto line BC, and define B₁ and C₁ cyclically. The triangle A₁B₁C₁ is the 2nd Hatzipolakis triangle. (Reference: X1119)

A-vertex coordinates:

Trilinears 0 : 1/(b*(a^2-b^2+c^2)*(a-b+c)^2) : 1/(c*(a^2+b^2-c^2)*(a+b-c)^2)

3rd Hatzipolakis

Let A'B'C' be the orthic triangle of ABC. Let A_b and A_c be the orthogonal projections of A' on AC and AB, respectively, and denote N_a the nine-point-center of AA_bA_c; build N_b and N_c cyclically. N_aN_bN_c is the 3rd Hatzipolakis triangle. (Reference: X17817)
A-vertex coordinates:

Trilinears 2*S^2*(10*R^2-2*SA-SB-SC)/a : b*(S^2+SA*SB) : c*(S^2+SA*SC)

Hatzipolakis-Moses

Let A'B'C' be the orthic triangle of ABC. Let A_B be the orthogonal projection of A' onto line AB, and define B_C and C_A cyclically. Let A_C be the orthogonal projection of A' onto line AC, and define B_A and C_B cyclically. Let A'_B be the reflection of A in A_B, and let A'_C be the reflection of A in A_C . Let N_A be the nine-point center of the triangle AA'_BA'_C, and define N_B and N_C cyclically. The triangle N_AN_BN_C, is the Hatzipolakis-Moses triangle. (Reference: X6145)
A-vertex coordinates:

Trilinears 2*S^2*(6*R^2-2*SA-SB-SC)/a : b*(S^2+SA*SB) : c*(S^2+SC*SA)

2nd Hatzipolakis-Moses

Let ABC be a triangle with orthocenter H and and A'B'C' its orthic triangle. Denote N_ab, N_ac the nine-points centers or X(5)-of-triangles AHB', AHC', respectively. Similarly define N_bc, N_ba and N_ca, N_cb.The triangle A"B"C" bounded by the lines N_abN_ac N_bcN_ba, N_caN_cb is the 2nd Hatzipolakis-Moses triangle. (Reference: Preamble just before X63656)
A-vertex coordinates:

Trilinears a*S^2*(SA+SW-8*R^2) : (S^4+SC*(2*SA^2*(SB+SC)+(4*SA+SC)*S^2))/b : (S^4+SB*(2*SA^2*(SB+SC)+(4*SA+SB)*S^2))/c

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 the hexyl triangle of ABC. (Reference: MathWorld -- Hexyl Triangle.)

Equivalences:

(anticomplementary)-of-(2nd circumperp), (anti-Euler)-of-(1st circumperp), (anti-X3-ABC reflections)-of-(6th mixtilinear), (excentral)-of-(ABC-X3 reflections), (Johnson)-of-(excentral), (6th mixtilinear)-of-(anti-Aquila), (Ursa-minor)-of-(2nd circumperp tangential), (Wasat)-of-(anti-Euler).

A-vertex coordinates:

Trilinears a^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_BC_B∩A_CB_C, and define B* and C* cyclically. The triangle A*B*C* is the Honsberger triangle. (Reference: X7670)

A-vertex coordinates:

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

Hung-Feuerbach

Let F_aF_bF_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 the Hung-Feuerbach triangle of 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. The inner-Hutson triangle is the triangle bounded by the lines L_A, L_B and L_C. (Reference: X363)

A-vertex coordinates:

Trilinears
-(2*δ²*sin(A/2)-y²*sin(B/2)-z²*sin(C/2)) :
(2*(c*z-δ²)*b*sin(A/2)-(a-c)*y²*sin(B/2)-b*z²*sin(C/2))/b :
(2*(b*y-δ²)*c*sin(A/2)-(a-b)*z²*sin(C/2)-c*y²*sin(B/2))/c

where x = (-a+b+c)/2, y = (a-b+c)/2, z = (a+b-c)/2 and δ² = (4*R*r+r²)/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. The outer-Hutson triangle is the triangle bounded by the lines M_A, M_B and M_C. (Reference: X363)

A-vertex coordinates:

Trilinears
-(2*δ²*sin(-A/2)-y²*sin(B/2)-z²*sin(C/2)) :
(2*(c*z-δ²)*b*sin(-A/2)-(a-c)*y²*sin(B/2)-b*z²*sin(C/2))/b :
(2*(b*y-δ²)*c*sin(-A/2)-(a-b)*z²*sin(C/2)-c*y²*sin(B/2))/c

where x = (-a+b+c)/2, y = (a-b+c)/2, z = (a+b-c)/2 and δ² = (4*R*r+r²)/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)

Equivalences:

(ABC-X3 reflections)-of-(intouch), (anti-Ara)-of-(excenters-reflections), (1st anti-circumperp)-of-(midarc), (2nd anti-circumperp-tangential)-of-(2nd tangential-midarc), (circumorthic)-of-(2nd midarc), (1st circumperp)-of-(2nd anti-circumperp-tangential), (2nd circumperp)-of-(Mandart-incircle), (Euler)-of-(Ursa-minor), (Mandart-incircle)-of-(tangential-midarc).

A-vertex coordinates:

Trilinears 4*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 the 1st 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 triangle of ABC. (Reference: Hyacinthos #24020)

Equivalences:

(Aries)-of-(anti-Ara).

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.

Equivalences:

anticomplement of (Gemini 16), complement of (Gemini 18).

A-vertex coordinates:

Trilinears 0 : 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 triangle of ABC. (Reference: X11034)

Equivalences:

(anti-Ara)-of-(2nd circumperp), (anti-X3-ABC reflections)-of-(intouch), (Johnson)-of-(inverse-in-incircle).

A-vertex coordinates:

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

infinite altitude

Let A', B', C' be the intersections of the line at infinity and the altitudes AX(4), BX(4), CX(4) of ABC, respectively. The triangle A'B'C' is the infinite altitude triangle of ABC. (Reference: Preamble just before X36436)

Equivalences:

isogonal of (ABC-X₃ reflections).

A-vertex coordinates:

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

intangents

In ABC, let t_a be the other than BC interior common tangent of the incircle and the A-excircle; define t_b and t_c cyclically. The triangle A'B'C' bounded by these lines is the intangents triangle.

Equivalences:

(anti-Hutson intouch)-of-(2nd anti-circumperp-tangential), (tangential)-of-(Mandart-incircle).

A-vertex coordinates:

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

intouch

The intouch triangle of ABC, also called the contact triangle, has as vertices the touchpoints of the incircle and the sidelines of ABC.

Equivalences:

anticomplement of (2nd Zaniah), complement of (inner-Conway).

(ABC-X3 reflections)-of-(Hutson intouch), (anti-Ara)-of-(excentral), (1st anti-circumperp)-of-(2nd midarc), (2nd anti-circumperp-tangential)-of-(tangential-midarc), (anticomplementary)-of-(inverse-in-incircle), (circumorthic)-of-(midarc), (1st circumperp)-of-(Mandart-incircle), (2nd circumperp)-of-(2nd anti-circumperp-tangential), (Hutson intouch)-of-(5th mixtilinear), (incircle-circles)-of-(Aquila), (Mandart-incircle)-of-(2nd tangential-midarc), (medial)-of-(Ursa-minor), (X3-ABC reflections)-of-(incircle-circles).

A-vertex coordinates:

Trilinears 0 : 1/(b*(a-b+c)) : 1/(c*(a+b-c))

inverse-in-Conway

Let A', B', C' be the inverses of A, B, C in the Conway circle. A'B'C' is the inverse-in-Conway triangle. (Reference: X10439)

Equivalences:

(medial)-of-(3rd Conway).

A-vertex coordinates:

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

inverse-in-excircles

The inverse-in-excircles triangle of ABC has as vertices the inverses of A,B,C in the A-, B-, C- excircle of ABC, respectively. (Reference: X8055)

A-vertex coordinates:

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

inverse-in-incircle

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

Equivalences:

(Johnson)-of-(incircle-circles), (medial)-of-(intouch).

A-vertex coordinates:

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

1st and 2nd isodynamic-Dao

Let P₁ = X(15) be the 1st isodynamic point of ABC. Circles (A, P₁), (B, P₁) and (C, P₁) intersect by pairs at P₁, A', B', C'. Then the triangle A'B'C' is equilateral and is named the 1st isodynamic-Dao equilateral triangle.

Let P₂ = X(16) be the 2nd isodynamic point of ABC. Circles (A, P₂), (B, P₂) and (C, P₂) intersect by pairs at P₂, A", B", C". Then the triangle A"B"C" is equilateral and is called the 2nd isodynamic-Dao equilateral triangle. (Reference: X16802)

Equivalences:

1st isodynamic-Dao = (anticomplementary)-of-(3rd isodynamic-Dao), (anti-Ehrmann-mid)-of-(3rd isodynamic-Dao), (3rd anti-Euler)-of-(3rd isodynamic-Dao), (anti-Mandart-incircle)-of-(3rd isodynamic-Dao), (Ehrmann-mid)-of-(15th Fermat-Dao), (3rd Euler)-of-(15th Fermat-Dao), (excentral)-of-(3rd isodynamic-Dao), (Mandart-incircle)-of-(15th Fermat-Dao), (reflection)-of-(3rd isodynamic-Dao), (tangential)-of-(3rd isodynamic-Dao).

2nd isodynamic-Dao = (anticomplementary)-of-(4th isodynamic-Dao), (anti-Ehrmann-mid)-of-(4th isodynamic-Dao), (3rd anti-Euler)-of-(4th isodynamic-Dao), (anti-Mandart-incircle)-of-(4th isodynamic-Dao), (Ehrmann-mid)-of-(16th Fermat-Dao), (3rd Euler)-of-(16th Fermat-Dao), (excentral)-of-(4th isodynamic-Dao), (Mandart-incircle)-of-(16th Fermat-Dao), (reflection)-of-(4th isodynamic-Dao), (tangential)-of-(4th isodynamic-Dao).

A-vertex coordinates:

1st isodynamic-Dao equilateral triangle:
Trilinears
-(SB+SC)*(sqrt(3)*SA+S)/a : (sqrt(3)*(S^2+SA*SC)+(SA+3*SC)*S)/b : (sqrt(3)*(S^2+SA*SB)+(SA+3*SB)*S)/c

2nd isodynamic-Dao equilateral triangle:
Trilinears
-(SB+SC)*(sqrt(3)*SA-S)/a : (sqrt(3)*(S^2+SA*SC)-(SA+3*SC)*S)/b : (sqrt(3)*(S^2+SA*SB)-(SA+3*SB)*S)/c

3rd and 4th isodynamic-Dao

Let A_oB_oC_o be the orthic triangle of ABC. Denote by A', B', C' the 1st isodynamic points X(15) of AB_oC_o, BC_oA_o and CA_oB_o, respectively, and A", B", C" the 2nd isodynamic points X(16) of AB_oC_o, BC_oA_o and CA_oB_o, respectively. Then the triangles A'B'C' and A"B"C" are equilateral and are the 3rd and 4th isodynamic-Dao of ABC, respectively. (Reference: preamble just before X31683)

Equivalences:

3rd isodynamic-Dao = (anti-Ara)-of-(15th Fermat-Dao), (2nd anti-Conway)-of-(15th Fermat-Dao), (6th anti-mixtilinear)-of-(15th Fermat-Dao), (Ehrmann-mid)-of-(1st isodynamic-Dao), (3rd Euler)-of-(1st isodynamic-Dao), (inverse-in-incircle)-of-(15th Fermat-Dao), (Mandart-incircle)-of-(1st isodynamic-Dao), (medial)-of-(1st isodynamic-Dao).

4th isodynamic-Dao = (anti-Ara)-of-(16th Fermat-Dao), (2nd anti-Conway)-of-(16th Fermat-Dao), (6th anti-mixtilinear)-of-(16th Fermat-Dao), (Ehrmann-mid)-of-(2nd isodynamic-Dao), (3rd Euler)-of-(2nd isodynamic-Dao), (inverse-in-incircle)-of-(16th Fermat-Dao), (Mandart-incircle)-of-(2nd isodynamic-Dao), (medial)-of-(2nd isodynamic-Dao).

A-vertex coordinates:

3rd isodynamic-Dao equilateral triangle:
Trilinears
(sqrt(3)*(S^2+SB*SC)+2*(SB+SC)*S)/(a*SA) : (sqrt(3)*SC+S)/b : (sqrt(3)*SB+S)/c

4th isodynamic-Dao equilateral triangle:
Trilinears
(sqrt(3)*(S^2+SB*SC)-2*(SB+SC)*S)/(a*SA) : (sqrt(3)*SC-S)/b : (sqrt(3)*SB-S)/c

J:

1st, 2nd and 3rd Jenkins

Let {a'} be the circle internally tangent to the A-excircle and externally tangent to the B- and C- excircles and build {b'}, {c'} cyclically. Let A', B', C' be the centers of these just defined circles. The triangle A'B'C' is the 1st Jenkins triangle of ABC. (Reference: "Hechos Geométricos en el Triángulo", by Angel Montesdeoca).

Notes: Circles {a'}, {b'}, {c'} are named the A-, B- and C- Jenkins circles. (Reference: Jenkins circles at "The Triangle Web", by Quim Castellsaguer). These circles concur at X(10).

Let A" be the intersection, other than X(10), of the B- and C- Jenkins circles and denote B", C" cyclically. The triangle A"B"C" is the 2nd Jenkins triangle of ABC. (Reference: "Hechos Geométricos en el Triángulo", by Angel Montesdeoca).

Let T_ab and T_ac be the touchpoints of the A-Jenkins circle with the B- and C- excircles, respectively. Denote (T_bc, T_ba) and (T_ca, T_cb) cyclically. The triangle bounded by the lines T_abT_ac, T_bcT_ba and T_caT_cb is named the 3rd Jenkins triangle of ABC.

A-vertex coordinates:

1st Jenkins triangle:

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

2nd Jenkins triangle:

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

3rd Jenkins triangle:

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

Jenkins-contact

Let T_a be the touchpoint of the A-Jenkins circle and the A-excircle and denote T_b, T_c cyclically. The triangle T_aT_bT_c is the Jenkins-contact triangle of ABC. (See Jenkins triangle). (Reference: "Hechos Geométricos en el Triángulo", by Angel Montesdeoca).

A-vertex coordinates:

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

Jenkins-tangential

Let τ_a be the tangent to the A-Jenkins circle at T_a (see Jenkins-contact triangle) and define τ_b and τ_c cyclically. The triangle bounded by these tangents is the Jenkins-tangential triangle of ABC. (References: "Hechos Geométricos en el Triángulo", by Angel Montesdeoca and X(37865)).

A-vertex coordinates:

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

Johnson

The vertices of the Johnson triangle are the centers of the Johnson circles J_a, J_b and J_c. (Reference: MathWorld -- Johnson Circles.)
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 as Carnot triangle.

Equivalences:

anticomplement of (anti-X3-ABC reflections), complement of (anti-Euler).

(anticomplementary)-of-(Euler), (anti-Ehrmann-mid)-of-(anti-X3-ABC reflections), (anti-Euler)-of-(medial), (3rd anti-Euler)-of-(4th Euler), (4th anti-Euler)-of-(3rd Euler), (anti-X3-ABC reflections)-of-(anticomplementary), (Ehrmann-mid)-of-(anti-Euler), (Euler)-of-(anti-Ehrmann-mid), (medial)-of-(X3-ABC reflections), (X3-ABC reflections)-of-(Ehrmann-mid).

Only for acute ABC: (2nd circumperp)-of-(Ehrmann-side), (excentral)-of-(2nd Euler), (hexyl)-of-(orthic), (incircle-circles)-of-(anti-inverse-in-incircle), (inverse-in-incircle)-of-(anti-incircle-circles).

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 (see Johnson 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. The inner-Johnson triangle is the triangle enclosed by the lines L_A, L_B and L_C. (Reference: X10522)

Equivalences:

(anti-Ursa minor)-of-(Ursa-major).

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 (see Johnson 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. The outer-Johnson triangle is 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 (reference: MathWorld -- Yff Circles.). 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 the 1st Johnson-Yff triangle are the other points of intersection of Y'_a, Y'_b and Y'_c. (Reference: X10523).

Equivalences:

(Johnson)-of-(inner-Yff).

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/(-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 (reference: MathWorld -- Yff Circles.). 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 the 2nd Johnson-Yff triangle are the other points of intersection of Y"_a, Y"_b and Y"_c. (Reference: X10523).

Equivalences:

(Johnson)-of-(outer-Yff).

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:

K798e and K798i

Let A_e, A_i be the points where the perpendicular bisector of BC is cut by the circle through the nine-point-center of ABC and centered at the midpoint of BC, with A_i closer to A than A_e. Build B_i, B_e, C_i, C_e cyclically. A_eB_eC_e and A_iB_iC_i are the K798e & K798i triangles, respectively. (References: CTC K798 and X(11230) & X(11231))

A-vertex coordinates of K798e:

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

A-vertex coordinates of K798i:

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

K1343

The cubic K1343 cuts each perpendicular bisectors of ABC in two points. Three of these points are the vertices of the Kosnita triangle, the other three are the vertices of the K1343 triangle.

Note: Let A*B*C* be the Ehrmann-side triangle of ABC and let A', B', C' be the orthocenters of A*BC, B*CA, C*AB, respectively. Then A'B'C' is the K1343 triangle.

A-vertex coordinates:

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

1st and 2nd Kenmotu-centers

Let A', B', C' be the centers of the inner-Kenmotu squares, as showed in MathWorld's Kenmotu Point. Triangle A'B'C' is named here the 1st Kenmotu-centers triangle of ABC. As there exists an outer version of these squares with centers A", B", C", the triangle A"B"C" is refered here as the 2nd Kenmotu-centers triangle of ABC.

Equivalences:

1st Kenmotu-centers = (anti-X3-ABC reflections)-of-(1st Kenmotu-free-vertices).

2nd Kenmotu-centers = (anti-X3-ABC reflections)-of-(2nd Kenmotu-free-vertices).

A-vertex coordinates:

1st triangle:

Trilinears (a^2+2*S)/a : b : c
2nd triangle:

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

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: MathWorld -- Kenmotu Point.). 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.

Equivalences:

(Kosnita)-of-(1st Kenmotu-free-vertices), (tangential)-of-(1st Kenmotu-centers).

A-vertex coordinates:

Trilinears a*(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: MathWorld -- Kenmotu Point.). 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.

Equivalences:

(Kosnita)-of-(2nd Kenmotu-free-vertices), (tangential)-of-(2nd Kenmotu-centers).

A-vertex coordinates:

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

1st & 2nd Kenmotu-free-vertices

Let K_a, K_b, K_c be the free vertices of the 1st(2nd) Kenmotu squares. Triangle K_aK_bK_c is the 1st(2nd) Kenmotu free vertices triangle. (Reference: X12375)

Equivalences:

1st Kenmotu-free-vertices = (X3-ABC reflections)-of-(1st Kenmotu-centers).

2nd Kenmotu-free-vertices = (X3-ABC reflections)-of-(2nd Kenmotu-centers).

A-vertex coordinates:

1st Kenmotu free vertices:

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

2nd Kenmotu free vertices:

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

Kosnita

The vertices of the Kosnita triangle are the circumcenters of the triangles BOC, COA, AOB, where O is the circumcenter of ABC. (Reference: X1658)

Equivalences:

(Ehrmann-vertex)-of-(Johnson), (tangential)-of-(anti-X3-ABC reflections), (Trinh)-of-(ABC-X3 reflections).

Only for acute ABC: (anti-Aquila)-of-(tangential).

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:

largest-circumscribed-equilateral

Considere all equilateral triangles A_eB_eC_e circumscribing ABC and such that A lies between B_e and C_e, B lies between C_e and A_e and C lies between A_e and B_e. The triangle defined here is the A_eB_eC_e with largest area. (Reference: Preamble just before X36761.)

This triangle is the antipedal triangle of X(13). The A-vertex is the antipode of X(13) in the circle {{X(13), B, C}}, and the other two vertices are found cyclically.

A-vertex coordinates:

Trilinears
(-6*sqrt(3)*S*a^2-3*(a^2+b^2+c^2)*a^2+2*(b^2-c^2)^2)/a :
((7*b^2+2*c^2)*a^2-3*b^4+5*b^2*c^2-2*c^4+2*sqrt(3)*S*(2*a^2+b^2))/b :
((2*b^2+7*c^2)*a^2-3*c^4+5*b^2*c^2-2*b^4+2*sqrt(3)*S*(2*a^2+c^2))/c

inner- & outer- Le Viet An

Let ABC be a triangle and BCA', CAB', ABC' equilateral triangles erected in/out - wardly of ABC. Let B_c, C_b be the circumcenters of CC'B and BB'C, respectively, and build C_a, A_c, A_b and B_a cyclically. Denote the circumcenters of B_cC_bA, C_aA_cB, A_bB_aC as O_a, O_b, O_c, respectively. Then, in each case, the triangle O_aO_bO_c is equilateral. These triangles are named the inner- and outer- Le Viet An triangles. (Reference: X14169)

Note: Both triangles are equilateral.

A-vertex coordinates:

inner-Le Viet An triangle:

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

outer-Le Viet An triangle:

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

Lemoine

The Lemoine triangle is the cevian triangle of X(598).

A-vertex coordinates:

Trilinears 0 : a*c/(2*a^2+2*c^2-b^2) : a*b/(2*a^2+2*b^2-c^2)

1st and 2nd Lemoine-Dao

Like in the construction of the 2nd Lemoine circle of a triangle ABC, let A_b, A_c be the points where the antiparallel to BC through X(6) cuts the sides AC and AB, respectively. Define B_c, B_a, C_a, C_b cyclically. Build equilateral triangles B_cC_bA', C_aA_cB' and A_bB_aC' all with the same orientation as ABC and build equilateral triangles B_cC_bA", C_aA_cB" and A_bB_aC", all with opposite orientation than ABC. Then A'B'C' and A"B"C" are both equilateral triangles and are named the 1st Lemoine-Dao equilateral triangle and the 2nd Lemoine-Dao equilateral triangle, respectively. (Reference: X16940)

A-vertex coordinates:

1st Lemoine-Dao equilateral triangle:
Trilinears
-(sqrt(3)*SA+2*S)*(SB+SC)/(a*SA) : (sqrt(3)*SC-S)/b : (sqrt(3)*SB-S)/c

2nd Lemoine-Dao equilateral triangle:
Trilinears
-(sqrt(3)*SA-2*S)*(SB+SC)/(a*SA) : (sqrt(3)*SC+S)/b : (sqrt(3)*SB+S)/c

Lucas triangles

Build outwards ABC the square BCC_AB_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'_BA'_CA"_BA"_C is a square named the A-Lucas square (and also the A-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 the A-, B-, C- Lucas circles and result to be pairwise tangents and simultaneously tangent internally to the circumcircle of ABC. The circle externally tangent to the Lucas circles is named the Lucas inner circle.

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

Denote:
A₀ = the center of the A-Lucas circle, and similarly B₀, C₀;
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;
A₂ = the center of the A-Lucas secundary central circle, and similarly B₂, C₂;
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 the Lucas 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 name Lucas with Lucas(-1); examples: A-Lucas(-1) square and A-Lucas(-1) circle instead of A-Lucas square and A-Lucas circle. Moreover, if the construction is generalized and BCC_AB_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 replacing Lucas by Lucas(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 and Lucas(-1) antipodal

Vertices: A_p, B_p and C_p. See Lucas triangles. (Reference: X6457)

A-vertex coordinates:

Lucas antipodal:
Trilinears cos(B)*cos(C)-sin(A) : -cos(B) : -cos(C)

Lucas(-1) antipodal:
Trilinears cos(B)*cos(C)+sin(A) : -cos(B) : -cos(C)

Lucas antipodal tangents and Lucas(-1) antipodal tangents

The tangents at A_p, B_p and C_p to the A-, B- and C-Lucas circles, respectively, bound the Lucas antipodal tangents triangle.

A-vertex coordinates:

Lucas antipodal tangents:
Trilinears (2*SA+SB+SC+S)*SA*a : (S^2+SB*SC+(2*SB+SA)*S)*b : (S^2+SB*SC+(2*SC+SA)*S)*c

Lucas(-1) antipodal tangents:
Trilinears (2*SA+SB+SC-S)*SA*a : (S^2+SB*SC-(2*SB+SA)*S)*b : (S^2+SB*SC-(2*SC+SA)*S)*c

Lucas Brocard and Lucas(-1) Brocard

The Lucas Brocard triangle is bounded by the radical axis of the Brocard circle and the Lucas circles. See Lucas triangles. (Reference: X6421)

A-vertex coordinates:

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

Lucas(-1) Brocard:
Trilinears -a*(a^2+2*S) : b*(a^2+c^2+S) : c*(a^2+b^2+S)

Lucas central and Lucas(-1) central

Vertices: A₀, B₀ and C₀. See Lucas triangles.

Equivalences:

Lucas central = (tangential)-of-(Lucas tangents).

Lucas(-1) central = (tangential)-of-(Lucas(-1) tangents).

A-vertex coordinates:

Lucas central:
Trilinears cos(A)+2*sin(A) : cos(B) : cos(C)

Lucas(-1) central:
Trilinears cos(A)-2*sin(A) : cos(B) : cos(C)

Lucas homothetic and Lucas(-1) homothetic

The Lucas homothetic triangle is enclosed by the lines A"_BA"_C, B"_CB"_A and C"_BC"_B. See Lucas triangles. (Reference: X493)

A-vertex coordinates:

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

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

Lucas inner and Lucas(-1) inner

Vertices: A_i, B_i and C_i. See Lucas triangles.

A-vertex coordinates:

Lucas inner:
Trilinears 2*cos(A)+3*sin(A)/2 : 2*cos(B)+sin(B) : 2*cos(C)+sin(C)

Lucas(-1) inner:
Trilinears 2*cos(A)-3*sin(A)/2 : 2*cos(B)-sin(B) : 2*cos(C)-sin(C)

Lucas inner tangential and Lucas(-1) 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)

Equivalences:

Lucas inner tangential = (tangential)-of-(Lucas inner).

Lucas(-1) inner tangential = (tangential)-of-(Lucas(-1) inner).

A-vertex coordinates:

Lucas inner tangential:
Trilinears 4*cos(A)+sin(A) : 4*cos(B)+3*sin(B) : 4*cos(C)+3*sin(C)

Lucas(-1) inner tangential:
Trilinears 4*cos(A)-sin(A) : 4*cos(B)-3*sin(B) : 4*cos(C)-3*sin(C)

Lucas reflection and Lucas(-1) reflection

Let L_a be the reflection of the line B₀C₀ in BC and define L_b, L_c cyclically (see Lucas triangles). The Lucas reflection triangle is the triangle limited by L_a, L_b, L_c. (Reference: X6401)

A-vertex coordinates:

Lucas reflection:
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:
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 and Lucas(-1) secondary central

Vertices: A₂, B₂ and C₂. See Lucas triangles. (Reference: X6199)

A-vertex coordinates:

Lucas secondary central:
Trilinears cos(A)-2*sin(A) : cos(B)+4*sin(B) : cos(C)+4*sin(C)

Lucas(-1) secondary central:
Trilinears cos(A)+2*sin(A) : cos(B)-4*sin(B) : cos(C)-4*sin(C)

Lucas 1st secondary tangents and Lucas(-1) 1st secondary tangents

Triangle enclosed by the lines B_2AC_2A, C_2BA_2B and A_2CB_2C (see Lucas triangles). (Reference: X6199)

A-vertex coordinates:

Lucas 1st secondary tangents:
Trilinears cos(A)-2*sin(A) : cos(B)+3*sin(B) : cos(C)+3*sin(C)

Lucas(-1) 1st secondary tangents:
Trilinears cos(A)+2*sin(A) : cos(B)-3*sin(B) : cos(C)-3*sin(C)

Lucas 2nd secondary tangents and Lucas(-1) 2nd secondary tangents

Triangle enclosed by the lines A_2BA_2C, B_2CB_2A and C_2AC_2B (see Lucas triangles). (Reference: X6199)

A-vertex coordinates:

Lucas 2nd secondary tangents
Trilinears cos(A)+6*sin(A) : cos(B)-sin(B) : cos(C)-sin(C)

Lucas(-1) 2nd secondary tangents
Trilinears cos(A)-6*sin(A) : cos(B)+sin(B) : cos(C)+sin(C)

Lucas tangents and Lucas(-1) tangents

Vertices: A_T, B_T and C_T. See Lucas triangles.

Equivalences:

Lucas(+1) tangents = isogonal of (2nd outer-Vecten).

Lucas(-1) tangents = isogonal of (2nd inner-Vecten).

A-vertex coordinates:

Lucas tangents:
Trilinears cos(A) : cos(B)+sin(B) : cos(C)+sin(C)

Lucas(-1) tangents:
Trilinears cos(A) : cos(B)-sin(B) : cos(C)-sin(C)

M:

MacBeath

The MacBeath triangle is the triangle whose vertices are the contact points of the MacBeath inconic with the reference triangle ABC. (MathWorld -- MacBeath Triangle.).

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:

Trilinears 0 : 1/(b^3*(a^2-b^2+c^2)) : 1/(c^3*(a^2+b^2-c^2))

Malfatti

The Malfatti triangle, or better named, the inner-Malfatti triangle, has as vertices the centers of the Malfatti circles.

Note: The Malfatti circles are three circles packed inside a triangle such that each is tangent to the other two and to two sides of the triangle. (Reference: MathWorld -- 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 named the outer-Malfatti circles. The centers of these circles are vertices of the outer-Malfatti triangle.

Equivalences:

inner-Malfatti = (tangential)-of-(inner-Malfatti-touchpoints).

outer-Malfatti = (tangential)-of-(outer-Malfatti-touchpoints).

A-vertex coordinates:

inner-Malfatti:
Trilinears ( 2*cos(B/2)*cos(C/2)+2*cos(B/2)+2*cos(C/2)-cos(A/2)+1)/(cos(A/2)+1) : 1 : 1

outer-Malfatti:
Trilinears (-2*cos(B/2)*cos(C/2)+2*cos(B/2)+2*cos(C/2)-cos(A/2)-1)/(cos(A/2)-1) : 1 : 1

Malfatti-touchpoints

Let A' be the touchpoint of the inner-B- and inner-C-Malfatti circles and define B', C' cyclically. The triangle A'B'C' is the inner-Malfatti-touchpoints triangle of ABC. (Reference: X46876)

Let A" be the touchpoint of the outer-B- and outer-C-Malfatti circles and define B", C" cyclically. The triangle A"B"C" is the outer-Malfatti-touchpoints triangle of ABC.

A-vertex coordinates:

inner-Malfatti-touchpoints:
Trilinears (1+cos(B/2))*(1+cos(C/2))/(1+cos(A/2)) : (1+cos(C/2))^2 : (1+cos(B/2))^2

outer-Malfatti-touchpoints:
Trilinears (1-cos(B/2))*(1-cos(C/2))/(1-cos(A/2)) : (1-cos(C/2))^2 : (1-cos(B/2))^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 triangle of 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 triangle of ABC.

Equivalences:

(ABC-X3 reflections)-of-(2nd anti-circumperp-tangential), (1st anti-circumperp)-of-(intouch), (2nd anti-circumperp-tangential)-of-(5th mixtilinear), (circumorthic)-of-(Hutson intouch), (orthic)-of-(Ursa-minor).

Only for acute ABC: (Hutson intouch)-of-(anti-tangential-midarc).

A-vertex coordinates:

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

McCay

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

Note: Let G be the centroid and A'B'C' the 2nd Brocard triangle of 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: MathWorld -- McCay Circles.)

A-vertex coordinates:

Trilinears a*(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 triangle is the cevian triangle of the centroid.

Equivalences:

complement of (ABC), anticomplement of (Gemini 110).

(ABC-X3 reflections)-of-(Euler), (1st anti-circumperp)-of-(3rd Euler), (anticomplementary)-of-(Gemini 110), (2nd anti-Conway)-of-(excentral), (circumorthic)-of-(4th Euler), (Euler)-of-(Johnson), (outer-Garcia)-of-(anti-Aquila), (Johnson)-of-(anti-X3-ABC reflections), (orthic)-of-(Wasat), (tangential)-of-(2nd Zaniah).

Only for acute ABC: (1st circumperp)-of-(orthic), (2nd circumperp)-of-(2nd Euler), (3rd Euler)-of-(anti-Wasat), (excentral)-of-(6th anti-mixtilinear), (inverse-in-incircle)-of-(tangential), (6th mixtilinear)-of-(submedial).

A-vertex coordinates:

Trilinears 0 : 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 triangle of ABC. (Reference: MathWorld -- Mid-Arc Triangle.)

Equivalences:

(ABC-X3 reflections)-of-(2nd midarc), (1st circumperp)-of-(Hutson intouch), (2nd circumperp)-of-(intouch), (4th Euler)-of-(Ursa-minor), (Hutson intouch)-of-(2nd tangential-midarc).

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 triangle of ABC.

Equivalences:

(ABC-X3 reflections)-of-(midarc), (1st circumperp)-of-(intouch), (2nd circumperp)-of-(Hutson intouch), (3rd Euler)-of-(Ursa-minor), (Hutson intouch)-of-(tangential-midarc).

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 triangle of ABC. (It is also known as the half-altitude triangle)

A-vertex coordinates:

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

mixtilinear

The mixtilinear triangle is the triangle connecting the centers of the mixtilinear incircles. It is alson known as the inner-mixtilinear triangle. (Reference: MathWorld -- Mixtilinear Triangle.)

Note: A circle that is simultaneously tangent to two sides of a triangle and internally tangent to its circumcircle is called a mixtilinear incircle. There are three mixtilinear incircles, one corresponding to each angle of the triangle (reference: MathWorld -- Mixtilinear Incircles.). Also, there are three circles that are each tangent to two sides of a triangle and externally tangent to its circumcircle; these are called the mixtilinear 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 excircles is the 2nd mixtilinear triangle. It is alson known as the outer-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 triangle has 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 triangle has as vertices the touchpoints of the circumcircle and the mixtilinear excircles. (See note in mixtilinear triangle). (Reference: Preamble just before X7955)

Note: The 4th mixtilinear triangle is the circumcevian triangle of X(55).

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 A₅=I_abI_ba∩I_acI_ca, B₅=I_bcI_cb∩I_baI_ab, C₅=I_caI_ac∩I_cbI_bc. A₅B₅C₅ is the 5th mixtilinear triangle. (See note in mixtilinear triangle).

Note: This triangle is also named Caelum triangle. (Reference: Preamble just before X7955)

Equivalences:

(anti-Hutson intouch)-of-(intouch), (anti-Mandart-incircle)-of-(2nd anti-circumperp-tangential), (2nd circumperp tangential)-of-(Mandart-incircle), (orthic)-of-(excenters-reflections), (tangential)-of-(Hutson intouch).

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. The 6th mixtilinear triangle is the triangle bounded by the lines E_abE_ac, E_bbE_ba and E_caE_cb. (See note in mixtilinear triangle). (Reference: Preamble just before X7955)

Equivalences:

(anticomplementary)-of-(excentral), (hexyl)-of-(Aquila), (X3-ABC reflections)-of-(hexyl).

A-vertex coordinates:

Trilinears a^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 A₇=E_abE_ba∩E_acE_ca, B₇=E_bcE_cb∩E_baE_ab, C₇=E_caE_ac∩E_cbE_bc. A₇B₇C₇ is the 7th 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)

8th mixtilinear

Let ω'_a, ω'_b and ω'_c be the A-, B- and C- mixtilinear incircles of ABC, and Ω_i the inner Apollonius circle of them. Denote as A', B', C' the touchpoints of Ω_i and ω'_a, ω'_b and ω'_c, respectively. The triangle A'B'C' is the 8th-mixtilinear triangle of ABC. (Reference: Preamble just before X44840)

A-vertex coordinates:

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

9th mixtilinear

Let ω"_a, ω"_b and ω"_c be the A-, B- and C- mixtilinear excircles of ABC, and Ω_o the outer Apollonius circle of them. Denote as A", B", C" the touchpoints of Ω_o and ω"_a, ω"_b and ω"_c, respectively. The triangle A"B"C" is the 9th-mixtilinear triangle of ABC. (Reference: Preamble just before X44840)

A-vertex coordinates:

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

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 the Hung-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 the Montesdeoca-Hung triangle. (Reference: X6042)

Note: The Apollonious circle is the circle internally tangent to all the excircles (MathWorld -- Apollonius Circle.)

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 A₁ = b_a∩c_a and similarly B₁ and C₁. The triangle A₁B₁C₁ is equilateral and is named the 1st Morley triangle. (MathWorld -- First Morley Triangle.)

Equivalences:

isogonal of (1st Morley-adjunct).

A-vertex coordinates:

Trilinears 1 : 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 A₂= b'_a∩c'_a and similarly B₂ and C₂. The triangle A₂B₂C₂ is equilateral and is named the 2nd Morley triangle. (MathWorld -- Second Morley Triangle.)

Note: Invert the senses of rotation if ABC is labeled clockwise.

Equivalences:

isogonal of (2nd Morley-adjunct).

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 A₃= b"_a∩c"_a and similarly B₃ and C₃. The triangle A₃B₃C₃ is equilateral and is named the 3rd Morley triangle. (MathWorld -- Third Morley Triangle.)

Note: Invert the senses of rotation if ABC is labeled clockwise.

Equivalences:

isogonal of (3rd Morley-adjunct).

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'₁B'₁C'₁ is the 1st Morley-adjunct triangle. (MathWorld -- First Morley Adjunct Triangle.)

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

Equivalences:

isogonal of (1st Morley).

A-vertex coordinates:

Trilinears 2 : 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'₂B'₂C'₂ is equilateral is the 2nd Morley-adjunct triangle. (MathWorld -- Second Morley Adjunct Triangle.)

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

Equivalences:

isogonal of (2nd Morley).

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'₃B'₃C'₃ is the 3rd Morley-adjunct triangle. (MathWorld -- Third Morley Adjunct Triangle.)

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

Equivalences:

isogonal of (3rd Morley).

A-vertex coordinates:

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

1st, 2nd and 3rd Morley-adjunct-midpoint

Let A_nB_nC_n be the n^th Morley-adjunct triangle (n=1,2,3). Let B_a and C_a be points on BC such that A_nB_aC_a is an equilateral triangle having the same orientation as ABC; define C_b and A_b cyclically, and define A_c and B_c cyclically. Let A"_n be the midpoint of A_b and A_c, and define B"_n and C"_n cyclically. Then A"_nB"_nC"_n is named the n^th Morley-adjunct-midpoint triangle. (Reference: X16839-to-X16841)

A-vertex coordinates:

Trilinears
(3*a*b*c*(2*a*y*z+b*x*z+c*x*y)+2*(2*sqrt(3)*S+3*SB+3*SC)*b*c*x+2*(3*SA+6*SC+sqrt(3)*S)*a*c*y+2*(6*SB+3*SA+sqrt(3)*S)*a*b*z+12*S^2+4*sqrt(3)*(SA+SW)*S+12*SB*SC)/(a*(3*b*c*x+6*SA-2*sqrt(3)*S)) : 2*b+x*c+z*a : 2*c+y*a+x*b

where x=sec((A-2*(n-1)*π)/3), y=sec((B-2*(n-1)*π)/3), z=sec((C-2*(n-1)*π)/3)

1st, 2nd and 3rd Morley-midpoint equilaterals

Let A_nB_nC_n be the n^th Morley triangle (n=1,2,3). Let B_a and C_a be points on BC such that A_nB_aC_a is an equilateral triangle having the same orientation as ABC; define C_b and A_b cyclically, and define A_c and B_c cyclically. Let A"_n be the midpoint of A_b and A_c, and define B"_n and C"_n cyclically. Then A"_nB"_nC"_n is an equilateral triangle and is named the n^th Morley-midpoint equilateral triangle. (Reference: X3602-to-X3604)

A-vertex coordinates:

Trilinears (12*a*b*c*(x*z*b+x*y*c+2*z*y*a)+2*(3*a^2+2*sqrt(3)*S)*b*c*x+2*(3*SA+6*SC+sqrt(3)*S)*a*c*y+2*(3*SA+6*SB+sqrt(3)*S)*a*b*z+3*S^2+sqrt(3)*(SA+SW)*S+3*SB*SC)/(a*(3*SA+6*x*b*c-sqrt(3)*S)) : 2*z*a+b+2*x*c : 2*y*a+2*x*b+c

where x=cos((A-2*(n-1)*π)/3), y=cos((B-2*(n-1)*π)/3), z=cos((C-2*(n-1)*π)/3)

Moses-Hung

Let H_aH_bH_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 the Moses-Hung triangle of 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

Moses-Miyamoto

Let γ be the incircle of ABC. Let γ_a be the circle through B and C and internally tangent to γ. Outside ABC, let Γ_a be the circle externally tangent to AB, AC, γ_a, and let A' be the center of Γa. Define B', C' cyclically. The triangle A'B'C' is the Moses-Miyamoto triangle. (Reference: X52804)

A-vertex coordinates:

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

Moses-Miyamoto-Apollonius, 1st and 2nd

In a scalene acute triangle ABC, let M_aM_bM_c be its medial triangle. Let Ω_a be the circle centered at M_a and passing through B and C. Inside ABC, let γ_a be the circle externally tangent to lines AB, AC and circle Ω_a. Define Ω_b, Ω_c γ_b and γ_c cyclically. Inside ABC, let γ be the circle internally tangent to Ω_a, Ω_b, Ω_c. Then there exists a circle Γ that is tangent to the four circles γ, γ_a, γ_b and γ_c.
Let A' be the touchpoint of Γ and γ_a, and define B' and C' cyclically. The triangle A'B'C' is the 1st Moses-Miyamoto-Apollonius triangle. (Reference: X52804)

In a scalene acute triangle ABC, let M_aM_bM_c be its medial triangle. Let Ω_a be the circle centered at M_a and passing through B and C. Inside ABC, let γ_a be the larger circle internally tangent to lines AB, AC and circle Ω_a. Define Ω_b, Ω_c γ_b and γ_c cyclically. Outside ABC, let γ be the circle internally tangent to Ω_a, Ω_b, Ω_c. Then there exists a circle Γ that is tangent to the four circles γ, γ_a, γ_b and γ_c.
Let A" be the touchpoint of Γ and γ_a, and define B" and C" cyclically. The triangle A"B"C" is the 2nd Moses-Miyamoto-Apollonius triangle. (Reference: X52804)

A-vertex coordinates:

1st Moses-Miyamoto-Apollonius triangle:

Trilinears -2/((a+b-c)*(a-b+c)-2*S) : 1/(b*(a-b+c)) : 1/(c*(a+b-c))
2nd Moses-Miyamoto-Apollonius triangle:

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

Moses-Soddy

Let A' be the pole of Soddy line in the Soddy A-circle, and define B' and C' cyclically. The triangle A'B'C' is the Moses-Soddy triangle. (Reference: X44311)

Note: Moses-Soddy triangle is the anticevian-triangle of X(514).

Equivalences:

complement of (Yff contact).

A-vertex coordinates:

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

Moses-Steiner osculatory

A' is the A-center of curvature of the Steiner circumellipse, and similarly B' and C'. A'B'C' is the Moses-Steiner osculatory triangle of ABC. (Reference: X34505)

A-vertex coordinates:

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

Moses-Steiner reflection

Let A', B', C' be the reflections of X(99) in the sides of Moses-Steiner osculatory triangle of ABC. A'B'C' is the Moses-Steiner reflection triangle of ABC. (Reference: X34505)

A-vertex coordinates:

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

N:

inner-Napoleon

The inner Napoleon triangle is 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. (MathWorld -- Inner Napoleon Triangle.)

Equivalences:

anticomplement of (2nd half-diamonds-central).

(anticomplementary)-of-(2nd half-diamonds-central), (anti-Ehrmann-mid)-of-(2nd half-diamonds-central), (3rd anti-Euler)-of-(2nd half-diamonds-central), (anti-Mandart-incircle)-of-(2nd half-diamonds-central), (excentral)-of-(2nd half-diamonds-central), (reflection)-of-(2nd half-diamonds-central), (tangential)-of-(2nd half-diamonds-central).

A-vertex coordinates:

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

outer-Napoleon

The outer Napoleon triangle is 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. (MathWorld -- Outer Napoleon Triangle.)

Equivalences:

anticomplement of (1st half-diamonds-central).

(anticomplementary)-of-(1st half-diamonds-central), (anti-Ehrmann-mid)-of-(1st half-diamonds-central), (3rd anti-Euler)-of-(1st half-diamonds-central), (anti-Mandart-incircle)-of-(1st half-diamonds-central), (excentral)-of-(1st half-diamonds-central), (reflection)-of-(1st half-diamonds-central), (tangential)-of-(1st half-diamonds-central).

A-vertex coordinates:

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

1st Neuberg

The triangle A₁B₁C₁ formed by joining the centers of the Neuberg circles is the 1st Neuberg triangle. (Weisstein, Eric W. "First Neuberg Triangle." From MathWorld)

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. (MathWorld -- Neuberg Circles.)

A-vertex coordinates:

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

2nd Neuberg

The triangle A₂B₂C₂ formed by joining the centers of the reflections of the Neuberg circles on the opposite sides of ABC is the 2nd Neuberg triangle. (MathWorld -- Second Neuberg Triangle.)

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).

Equivalences:

anticomplement of (6th anti-mixtilinear), complement of (1st anti-circumperp).

(ABC-X3 reflections)-of-(2nd Euler), (1st anti-circumperp)-of-(medial), (anticomplementary)-of-(2nd anti-Conway), (6th anti-mixtilinear)-of-(anticomplementary), (circumorthic)-of-(Euler), (Ehrmann-side)-of-(Ehrmann-mid), (2nd Euler)-of-(Johnson), (medial)-of-(anti-Wasat), (submedial)-of-(Gemini 111), (tangential)-of-(anti-Ara).

Only for acute ABC: (anti-Aquila)-of-(circumorthic), (Mandart-incircle)-of-(anti-Ursa minor).

A-vertex coordinates:

Trilinears 0 : 1/(b*(a^2-b^2+c^2)) : 1/(c*(a^2+b^2-c^2))

orthic axes

Let L_A be the orthic axis of triangle BCX(4), and define L_B, L_C cyclically. The triangle bounded by these lines is the orthic axes triangle. (Reference: X2501)
The orthic axes triangle is also the cevian triangle of X(4) with respect to the orthic triangle.

A-vertex coordinates:

Trilinears 2*SB*SC/a : SA*SC/b : SA*SB/c

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:

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

orthocentroidal-isogonic

Let A'B'C' be the orthocentroidal triangle of ABC and A", B", C" the isogonal conjugates of A', B', C' with respect to ABC, respectively. A"B"C" is the orthocentroidal-isogonic triangle of ABC.

A-vertex coordinates:

Trilinears 2*SB*SC/a : b*SB : c*SC

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:

Trilinears 2*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:

Paasche-Hutson

Let p_a be the parabola with focus A and directrix BC. Let B_a be the point, closer to C, at which p_a cuts AC and let C_a be the point, closer to B, at which p_a cuts AB. Build C_b, A_b and A_c, B_c cyclically. Points A_b, A_c, B_c, B_a, C_a, C_b lie on an ellipse named the Paasche ellipse.

Let A_H = A_bC_b∩A_cB_c, B_H = B_cA_c∩B_aC_a, C_H = C_aB_a∩C_bA_b. A_HB_HC_H is the Paasche-Hutson triangle. (Reference: X(37994))

A-vertex coordinates:

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

1st Pamfilos-Zhou

The Pamfilos-Zhou A-rectangle R_A = AA_BA_AA_C is the rectangle of maximal area such that A is a vertex of R_A, B lies on the line A_AA_C, and C lies on the line A_AA_B. The Pamfilos-Zhou B- and C-rectangles, R_B and R_C, are defined cyclically.

Let A' = B_CB_A∩C_AC_B, and define B' and C' cyclically. The 1st Pamfilos-Zhou triangle 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_BA_AA_C is the rectangle of maximal area such that A is a vertex of R_A, B lies on the line A_AA_C, and C lies on the line A_AA_B. The Pamfilos-Zhou B- and C-rectangles, R_B and R_C, are defined cyclically.

Let A" = C_AA_C∩A_BB_A, and define B" and C" cyclically. The 2nd Pamfilos-Zhou triangle 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 A₁ be the intersection, other than X(111), of the Parry circle and the line AX(111), and define B₁ and C₁ cyclically. The triangle A₁B₁C₁ is the 1st Parry triangle. (Reference: Preamble just before X9123)

Note: The Parry circle is the circle passing through the isodynamic points X(15) and X(16) and the centroid X(2) of a triangle.

Equivalences:

(ABC-X3 reflections)-of-(2nd Parry).

A-vertex coordinates:

Trilinears 3*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 A₂ be the intersection, other than X(110), of the Parry circle and line AX(110), and define B₂ and C₂ cyclically. The triangle A₂B₂C₂ is the 2nd Parry triangle. (Reference: Preamble just before X9123)

Note: The Parry circle is the circle passing through the isodynamic points X(15) and X(16) and the centroid X(2) of a triangle.

Equivalences:

(ABC-X3 reflections)-of-(1st Parry).

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 A₃ be the intersection, other than X(2), of the Parry circle and the A-median, and define B₃ and C₃ cyclically. The triangle A₃B₃C₃, is the 3rd Parry triangle. (Reference: Preamble just before X9123)

Note: The Parry circle is 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*(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)

1st and 2nd Pavlov

In a triangle ABC, let A'B'C' be its incentral triangle and I its incenter. Denote with A_b the projection of A' upon BI, and with A_c the projection of A' upon CI. Similarly define B_c, C_a and B_a, C_b. Lines B_cC_b, C_aA_c, and A_bB_a form a triangle A₁B₁C₁, called the 1st Pavlov triangle. Lines A_bA_c, B_cB_a, and C_aC_b bound the triangle A₂B₂C₂, named here the 2nd Pavlov triangle. (Reference: Ivan Pavlov, preamble just before X(63254).)

A-vertex coordinates:

1st Pavlov triangle:

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

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

Pelletier

The Pelletier triangle is the vertex triangle of ABC and the intangets triangle. Reference: TCCT, p. 162.

Note: The vertex triangle A*B*C* of triangles T₁=A₁B₁C₁ and T₂=A₂B₂C₂ has vertex A*=B₁B₂∩C₁C₂, and B*, C* are defined cyclically.

Note: The Pelletier triangle is the anticevian triangle of X(650) and, as an anticevian triangle, it is the tangential triangle of the circumconic with perspector X(650), i.e., it is the tangential triangle of the Feuerbach circum-hyperbola.

A-vertex coordinates:

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

1st Przybyłowski-Bollin

Let D₁ be the 1st isodynamic center X(15) of ABC and I'_a, I'_b and I'_c the incenters of BCD₁, CAD₁ and ABD₁, respectively. The triangle I'_aI'_bI'_c is the 1st Przybyłowski-Bollin triangle. (Reference: Preamble just before X11752)

Note: This triangle is denoted as AiBiCi in ETC.

A-vertex coordinates:

Trilinears a*(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 D₁ be the 1st isodynamic center X(15) of ABC and J'_a the excenter of BCD₁ opposed to D₁. Define J'_b and J'_c similarly. The triangle J'_aJ'_bJ'_c is the 2nd Przybyłowski-Bollin triangle. (Reference: Preamble just before X11752)

Note: This triangle is denoted as AaBbCb in ETC.

A-vertex coordinates:

Trilinears a*(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 D₂ be the 2nd isodynamic center X(16) of ABC and I"_a, I"_b and I"_c the incenters of BCD₂, CAD₂ and ABD₂, respectively. The triangle I"_aI"_bI"_c is the 3rd Przybyłowski-Bollin triangle. (Reference: Preamble just before X11752)

Note: This triangle is denoted as (AiBiCi)* in ETC.

A-vertex coordinates:

Trilinears a*(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 D₂ be the 2nd isodynamic center X(16) of ABC and J"_a the excenter of BCD₂ opposed to D₂. Define J"_b and J"_c similarly. The triangle J"_aJ"_bJ"_c is the 4th Przybyłowski-Bollin triangle. (Reference: Preamble just before X11752)

Note: This triangle is denoted as (AaBbCb)* in ETC.

A-vertex coordinates:

Trilinears a*(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 triangle of ABC.

A-vertex coordinates:

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

Roussel (equilateral)

If the angular trisectors of a triangle are produced to the circumcircle, then the chords of adjacent trisectors bound 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 & 2nd Savin

In a triangle ABC with incenter I, let A', B', C' be the orthogonal projections of I in BC, CA and AB, respectively. Then the line B'C', the perpendicular to BC through A' and the A-median of ABC concur in a point A₁ and points B₁ and C₁ are defined cyclically. The triangle A₁B₁C₁ is the 1st Savin triangle of ABC. (Reference: Preamble just before X44301)

No geometric construction was given for 2nd Savin triangle.

A-vertex coordinates:

1st triangle:

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

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

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_cC_b and similarly O'_b and O'_c. The triangle O'_aO'_bO'_c is the 1st Schiffler triangle. (Reference: X6595)

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)

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_cC_b and similarly O"_b and O"_c. The triangle O"_aO"_bO"_c is the 2nd 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 triangle of ABC.

Note: The Schroeter triangle is the anticevian triangle of X(523).

Equivalences:

complement of (Steiner).

A-vertex coordinates:

Trilinears 1/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

Soddy

Let A_oB_oC_o, A_iB_iC_i be the outer- and inner-Soddy triangles of ABC. Then B_oC_oC_iB_i are concyclic and similarly the other quartets of vertices. The centers of the circles circumscribing these quartets are the vertices of the Soddy triangle. (Reference: Preamble before X(31528))

Note: The Soddy triangle is the anticevian triangle of X(7).

A-vertex coordinates:

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

inner-Soddy

Given a triangle ABC with inner Soddy center S', the inner Soddy triangle is 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. (MathWorld -- Soddy Triangles.)

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. (MathWorld -- Soddy Circles.)

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 triangle is 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. (MathWorld -- Soddy Triangles.)

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. (MathWorld -- Soddy Circles.)

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))

2nd inner-Soddy and 2nd outer-Soddy

Let A'B'C' be the intouch triangle of ABC, A_oB_oC_o the outer-Soddy triangle and A_iB_iC_i the inner-Soddy triangles of ABC. Then:

B'C'C_iB_i are concyclic and similarly the other quartets of vertices. The centers of the circles circumscribing these quartets are the vertices of the 2nd inner-Soddy triangle.

B'C'C_oB_o are concyclic and similarly the other quartets of vertices. The centers of the circles circumscribing these quartets are the vertices of the 2nd outer-Soddy triangle.

(Reference: Preamble before X(31528))

A-vertex coordinates:

2nd inner-Soddy triangle:

Trilinears ((-a+b+c)*a+2*S)/(a*(-a+b+c)) : 1 : 1
2nd outer-Soddy triangle:

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

inner-squares

Build outwards ABC the square BCC_AB_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'_BA'_CA"_BA"_C is a square named the A-inner-inscribed-square (and also the A-Lucas square). The B- and C- inner-inscribed-squares can be built similarly. The centers of these squares are the vertices of the inner-squares triangles. (MathWorld -- Inner Inscribed Squares Triangle.)

A-vertex coordinates:

Trilinears 2*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_AB_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'_BA'_CA"_BA"_C is a square named the A-outer-inscribed-square (and also the A-Lucas(-1) square). The B- and C- outer-inscribed-squares can be built similarly. The centers of these squares are the vertices of the outer-squares triangles. (MathWorld -- Outer Inscribed Squares Triangle.)

A-vertex coordinates:

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

Stammler

The Stammler triangle is the triangle formed by the centers of the Stammler circles. It is an equilateral triangle. (MathWorld -- MathWorld -- Stammler Triangle.)

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. (MathWorld -- Stammler Circles.)

Equivalences:

(anticomplementary)-of-(circumtangential), (anti-Ehrmann-mid)-of-(circumtangential), (anti-Euler)-of-(circumnormal), (3rd anti-Euler)-of-(circumtangential), (4th anti-Euler)-of-(circumnormal), (anti-Hutson intouch)-of-(circumnormal), (anti-Mandart-incircle)-of-(circumtangential), (Aquila)-of-(circumnormal), (2nd circumperp tangential)-of-(circumnormal), (excentral)-of-(circumtangential), (hexyl)-of-(circumnormal), (reflection)-of-(circumtangential), (tangential)-of-(circumtangential), (X3-ABC reflections)-of-(circumnormal).

A-vertex coordinates:

Trilinears cos(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 triangle is the cevian triangle of the Steiner point X(99). (MathWorld -- Steiner Triangle.)

Equivalences:

anticomplement of (Schröeter).

A-vertex coordinates:

Trilinears 0 : (b^2-c^2)*(a^2-b^2)/b : (c^2-b^2)*(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_AO_BO_C is named the submedial triangle. (Reference: X9813)

Equivalences:

complement of (6th anti-mixtilinear).

(6th anti-mixtilinear)-of-(medial), (orthic)-of-(Gemini 110).

A-vertex coordinates:

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

symmedial

The symmedial triangle is the cevian triangle of the symmedian point X(6).

A-vertex coordinates:

Trilinears 0 : 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 triangle is 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.

Equivalences:

isogonal of (anticomplementary), anticomplement of (anti-Ursa minor), complement of (anti-inverse-in-incircle).

(anti-Hutson intouch)-of-(ABC-X3 reflections), (anti-incircle-circles)-of-(anti-X3-ABC reflections), (anti-inverse-in-incircle)-of-(medial), (anti-Ursa minor)-of-(anticomplementary), (Ehrmann-vertex)-of-(anti-Ehrmann-mid), (Kosnita)-of-(X3-ABC reflections), (medial)-of-(1st excosine), (orthic)-of-(Ara).

Only for acute ABC: (anti-Aquila)-of-(anti-incircle-circles), (anti-Mandart-incircle)-of-(1st anti-circumperp), (Aquila)-of-(Kosnita), (2nd circumperp tangential)-of-(circumorthic).

Note: The tangential triangle is the anticevian triangle of X(6).

A-vertex coordinates:

Trilinears -a : b : c

(1st) tangential-midarc

The tangential mid-arc triangle 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. (MathWorld -- Tangential Mid-Arc Triangle.)

Equivalences:

(anti-Hutson intouch)-of-(2nd midarc), (anti-Mandart-incircle)-of-(Hutson intouch), (2nd circumperp tangential)-of-(intouch), (tangential)-of-(midarc).

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-arc triangle 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)

Equivalences:

(anti-Hutson intouch)-of-(midarc), (anti-Mandart-incircle)-of-(intouch), (2nd circumperp tangential)-of-(Hutson intouch), (tangential)-of-(2nd midarc).

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))

Thomson

The Thomson cubic K002 cuts the circumcircle of a triangle ABC in A, B, C and other three points A', B', C'. Let A' be the point on the branch of K002 passing through A and choose B' and C' similarly. A'B'C' is the Thomson triangle.

Algebraic coordinates of vertices of Thomson triangle are very complicated to be written here.

Some other used triangles related to Thomson triangles are: Thomson-anticomplementary, Thomson-excentral, Thomson-medial and Thomson-orthic. Of course, algebraic expressions for the vertices of these triangles are also very complicated.

Numeric coordinates with respect to ETC's 6-9-13 triangle:

Thomson:

Trilinears
{{3.87136295325922, -1.48498096032229, 2.8819453221583},
{-0.629263011432855, 2.55535829974244, 2.16199935659095},
{12.9519946390386, 12.4724370423599, -10.9719433039008}}
Thomson-anticomplementary:

Trilinears
{{8.45136867434929, 16.5127763024386, -11.6918892694805},
{17.452620603748, 8.43209778230062, -10.2519973383382},
{-9.70989469721579, -11.4020597029505, 16.0158879826651}}
Thomson-excentral:

Trilinears
{{5.3112525691825, 19.0782363806524, -12.0186926597053},
{20.7720702849278, 5.71159495994938, -9.90062600648572},
{-3.15806347765949, -4.47821668670003, 8.19853610085032}}
Thomson-medial:

Trilinears
{{6.16136581380771, 7.51389767105683, -4.40497197365852},
{8.41167879615739, 5.49372804102233, -4.04499899087295},
{1.62104997091644, 0.535188669709552, 2.52197233937788}}
Thomson-orthic:

Trilinears
{{1.67590632156687, 4.23860048384595, -0.0672464630167511},
{5.51552805619645, 1.04218899889407, 0.373520687505509},
{1.53291973629658, 0.614305674705425, 2.50787452112836}}

inner tri-equilateral

In a triangle ABC, let's inscribe three congruent equilateral triangles PA_bA_c, PB_cB_a and PC_aC_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 the A-, B-, C- inner/outer equilateral triangles, respectively.

For the inner-equilateral triangles, the inner tri-equilateral triangle A_iB_iC_i is defined as the triangle bounded by the lines A_bA_c, B_cB_a and C_aC_b. (Reference: Preamble above X10631)

A-vertex coordinates:

Trilinears a*(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_bA_c, PB_cB_a and PC_aC_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 the A-, B-, C- inner/outer equilateral triangles, respectively.

For the outer-equilateral triangles, the outer tri-equilateral triangle A_oB_oC_o is defined as the triangle bounded by the lines A_bA_c, B_cB_a and C_aC_b. (Reference: Preamble above X10631)

A-vertex coordinates:

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

tri-squares

Inscribe three squares into a triangle ABC such that each square has two vertices on two distinct sides of ABC and the other vertices of the three squares coincide at the vertices of another triangle A'B'C'. See this figure. (Reference: Preamble above X13637).

There are four triangles A'B'C' for this construction. These are named the tri-squares triangles of ABC.

A-vertex coordinates:

1st tri-square triangle:

Trilinears 2*S/a : (a^2+3*b^2-c^2+2*S)/b : (a^2-b^2+3*c^2+2*S)/c
2nd tri-square triangle:

Trilinears -2*S/a : (a^2+3*b^2-c^2-2*S)/b : (a^2-b^2+3*c^2-2*S)/c
3rd tri-square triangle:

Trilinears 4*(a^2+S)/(a*(b^2+c^2-a^2+2*S)) : 1/b : 1/c
4th tri-square triangle:

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

tri-squares-central

The tri-squares-central triangles are the triangles whose vertices are the centers of the squares built in the tri-squares triangles. (Reference: Preamble above X13637).

A-vertex coordinates:

1st tri-square-central triangle:

Trilinears 2*(3*a^2+4*S)/a : (3*b^2+c^2-a^2+2*S)/b : (b^2+3*c^2-a^2+2*S)/c
2nd tri-square-central triangle:

Trilinears 2*(3*a^2-4*S)/a : (3*b^2+c^2-a^2-2*S)/b : (b^2+3*c^2-a^2-2*S)/c
3rd tri-square-central triangle:

Trilinears (a^2+2*S)/a : (b^2+S)/b : (c^2+S)/c
4th tri-square-central triangle:

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

Trinh

There is a unique equilateral triangle AA₁A₂ inscribed in the circumcircle of triangle ABC, where, for concreteness, the labels are fixed so that AA₁A₂ has the same orientation as ABC. Let BB₁B₂ and CC₁C₂ be the corresponding equilateral triangles. Let A' = B₁B₂∩C₁C₂, B' = C₁C₂∩A₁A₂ and C' = A₁A₂∩B₁B₂. The triangle A'B'C' is named the Trinh triangle of ABC. (Reference: X7688)

Equivalences:

(anti-Hutson intouch)-of-(anti-X3-ABC reflections), (Kosnita)-of-(ABC-X3 reflections).

Only for acute ABC: (anti-Aquila)-of-(anti-Hutson intouch).

A-vertex coordinates:

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

U:

Ursa-major

In a triangle ABC, the common internal tangents of the incircle and the A-excircle touch them in four concyclic points. Let {a'} be the circle through these touchpoints and denote a" the radical axis of {a'} and the A-excircle. Build b" and c" cyclically. The triangle bounded by a", b" and c" is the Ursa-major triangle of ABC. (Reference: Preamble just before X17603)

Equivalences:

(Ursa-minor)-of-(inner-Johnson).

A-vertex coordinates:

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

Ursa-minor

In a triangle ABC, the common internal tangents of the incircle and the A-excircle touch them in four concyclic points. Let {a'} be the circle through these touchpoints and denote a" the radical axis of {a'} and the incircle. Build b" and c" cyclically. The triangle bounded by a", b" and c" is the Ursa-minor triangle of ABC. (Reference: Preamble just before X17603)

Equivalences:

(anticomplementary)-of-(intouch), (anti-Euler)-of-(Hutson intouch), (3rd anti-Euler)-of-(2nd midarc), (4th anti-Euler)-of-(midarc), (excentral)-of-(Mandart-incircle), (hexyl)-of-(2nd anti-circumperp-tangential).

A-vertex coordinates:

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

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.

Equivalences:

complement of (2nd half-squares).

(medial)-of-(2nd half-squares).

A-vertex coordinates:

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

2nd and 3rd inner-Vecten

Given a triangle ABC, build the square σ_a = AA_bA_aA_c, with the same orientation as ABC, and such that B and C lie on the lines A_aA_b and A_aA_c, respectively. Define σ_b = BB_cB_bB_a and σ_c = CC_aC_cC_b cyclically. Let A_o, B_o, C_o be the centers of these squares.

A_aB_bC_c and A_oB_oC_o are the 2nd inner-Vecten and 3rd inner-Vecten of ABC, respectively. (Reference: Preamble just before X32488).

Equivalences:

2nd inner-Vecten = isogonal of (Lucas(-1) tangents).

3rd inner-Vecten = (6th anti-mixtilinear)-of-(2nd half-squares), (orthic)-of-(inner-Vecten).

A-vertex coordinates:

2nd inner-Vecten triangle:

Trilinears 1/(a*SA) : 1/(b*(SB-S)) : 1/(c*(SC-S))
3rd inner-Vecten:

Trilinears (3*S^2+SB*SC-2*S*SW)/(a*SA) : (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.

Equivalences:

complement of (1st half-squares).

(excentral)-of-(3rd outer-Vecten), (medial)-of-(1st half-squares).

A-vertex coordinates:

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

2nd and 3rd outer-Vecten

Given a triangle ABC, build the square σ_a = AA_bA_aA_c, with opposite orientation than ABC, and such that B and C lie on the lines A_aA_b and A_aA_c, respectively. Define σ_b = BB_cB_bB_a and σ_c = CC_aC_cC_b cyclically. Let A_o, B_o, C_o be the centers of these squares.

A_aB_bC_c and A_oB_oC_o are the 2nd outer-Vecten and 3rd outer-Vecten of ABC, respectively. (Reference: Preamble just before X32488).

Equivalences:

2nd outer-Vecten = isogonal of (Lucas(+1) tangents).

2nd outer-Vecten = (6th anti-mixtilinear)-of-(1st half-squares), (orthic)-of-(outer-Vecten).

A-vertex coordinates:

2nd outer-Vecten triangle:

Trilinears 1/(a*SA) : 1/(b*(SB+S)) : 1/(c*(SC+S))
3rd outer-Vecten:

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

Vijay 1st to 7th

In the plane of a triangle ABC, let A'B'C' = medial triangle and A"B"C" = orthic triangle, and let E_a denote the ellipse that passes through A and has foci B' and C'. Define ellipses E_b and E_c cyclically. Let A_b = the point, other than A, in E_a ∩ AB, and define B_c and C_a cyclically and let A_c = the point, other than A, in E_a ∩ AC, and define B_a and C_b cyclically.

Triangles 1st to 7th Vijay (A₁B₁C₁ to A₇B₇C₇) are defined as follows: (Reference: Preamble just before X44733).

A₁ = A_bC_b ∩ A_cB_c, and define B₁ and C₁ cyclically;

A₂ = B_aB_c ∩ C_aC_b, and define B₂ and C₂ cyclically;

A₃ = A_bB_c ∩ A_cC_b, and define B₃ and C₃ cyclically;

A₄ = A_bC_a ∩ A_cB_a, and define B₄ and C₄ cyclically;

A₅ = the point in E_b ∩ E_c closer to A, and define B₅ and C₅ cyclically;

A₆ = the point in E_b ∩ E_c farthest from A, and define B₆ and C₆ cyclically;

A₇ = BC ∩ A₅A₆, and define B₇ and C₇ cyclically.

A-vertex coordinates:

1st triangle:

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

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

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

Trilinears -(a^7-(b+c)*a^6-(b-c)^2*a^5+(b^2-c^2)*(b-c)*a^4-((b^2-c^2)^2-4*b^2*c^2)*a^3+((b^2-c^2)^2-4*b^2*c^2)*(b+c)*a^2+
(b^4+c^4-2*(2*b^2+3*b*c+2*c^2)*b*c)*(b+c)^2*a-(b^4+c^4-2*(2*b-c)*(b-2*c)*b*c)*(b+c)^3)/(2*a*(b+c)*(-a^2+b^2+c^2)) :
(a^3-(b+c)*a^2+(b^2+2*b*c-c^2)*a+(b+c)*(3*b^2-2*b*c+c^2))*c/b :
(a^3-(b+c)*a^2-(b^2-2*b*c-c^2)*a+(b+c)*(b^2-2*b*c+3*c^2))*b/c
5th triangle:

Trilinears (a+b+c)*(a^2-b^2+c^2)*(a^2+b^2-c^2)/a :
(a^2+b^2-c^2)*((2*a+2*b)*S-c*(a+b+c)*(a+b-c))/b :
(a^2-b^2+c^2)*((2*a+2*c)*S-b*(a+b+c)*(a-b+c))/c
6th triangle:

Trilinears -(a+b+c)*(a^2-b^2+c^2)*(a^2+b^2-c^2)/a :
(a^2+b^2-c^2)*((2*a+2*b)*S+c*(a+b+c)*(a+b-c))/b :
(a^2-b^2+c^2)*((2*a+2*c)*S+b*(a+b+c)*(a-b+c))/c
7th triangle:

Trilinears 0 : (a^2+b^2-c^2)*(a+b)/b : (a^2-b^2+c^2)*(a+c)/c

Vijay-Paasche-Hutson (without index)

See Vijay-Paasche-polar.

Vijay-Paasche-Hutson triangles 1 to 31

See here.

Vijay-Paasche-midpoints

The medial triangle of the 1st Vijay-Paasche-Hutson triangle is the Vijay-Paasche-midpoints triangle. (Reference: X(37881))

A-vertex coordinates:

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

Vijay-Paasche-polar

Let p_a be the parabola with focus A and directrix BC. Let B_a be the point, closer to C, at which p_a cuts AC and let C_a be the point, closer to B, at which p_a cuts AB. Build C_b, A_b and A_c, B_c cyclically. Points A_b, A_c, B_c, B_a, C_a, C_b lie on an ellipse named the Paasche ellipse.

Let A_P be the pole of A with respect to the Paasche ellipse and build B_P, C_P cyclically. A_PB_PC_P is the Vijay-Paasche-polar triangle. (Reference: X(37994))

Note: In some ETC centers ETC, this triangle is refered as Vijay-Paasche-Hutson (without index). In other centers it is named Vijay-Paasche-tangents.

A-vertex coordinates:

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

Vijay-Paasche-tangents

See Vijay-Paasche-polar.

Vu-Dao-isodynamic equilateral

Let A₀B₀C₀ be the orthic triangle of ABC. Let A₁₅ = X(15)-of-AB₀C₀, and define B₁₅ and C₁₅ cyclically, so that A₁₅B₁₅C₁₅ is the 3rd isodynamic-Dao equilateral triangle of ABC.

Let A'₁₅ be the point, other than A, in which the line AA₁₅ meets the circle {{A,B₀,C₀}}, and define B'₁₅ and C'₁₅ cyclically. The triangle A'₁₅B'₁₅C'₁₅ is the Vu-Dao-X(15)-isodynamic equilateral triangle.

If X(15) is replaced by X(16) in the above construction, the resulting triangle, A'₁₆B'₁₆C'₁₆, is the Vu-Dao-X(16)-isodynamic equilateral triangle. (Reference: X(5478))

A-vertex coordinates:

Vu-Dao-X(15)-isodynamic equilateral triangle
Trilinears ((S^2+3*SB*SC)*a^2-4*sqrt(3)*S*SB*SC)/(4*a*SA*(sqrt(3)*a^2-2*S)) : (sqrt(3)*SC-S)/b : (sqrt(3)*SB-S)/c
Vu-Dao-X(16)-isodynamic equilateral triangle
Trilinears ((S^2+3*SB*SC)*a^2+4*sqrt(3)*S*SB*SC)/(4*a*SA*(sqrt(3)*a^2+2*S)) : (sqrt(3)*SC+S)/b : (sqrt(3)*SB+S)/c

W:

Walsmith

The Walsmith triangle has A-vertex the intersection of the A-symmedian of ABC and the perpendicular at X(125) to BC. (Reference: Bernard Gibert cubic K1091)

A-vertex coordinates:

Trilinears -(2*a^6-(3*b^4-4*b^2*c^2+3*c^4)*a^2+(b^4-c^4)*(b^2-c^2))/(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)/a : b : c

Wasat

Let ABC be a triangle, A'B'C' its incentral triangle and (I) its incircle. Denote (O_a) the circle with diameter AA' and r_a the radical axis of (O_a) and (I); build (O_b), (O_c) and r_b, r_c cyclically. The triangle A*B*C* bounded by r_a, r_b, r_c is the Wasat triangle of ABC. (Reference: Preamble just before X21616)

Equivalences:

complement of (excentral).

(anticomplementary)-of-(3rd Euler), (anti-Euler)-of-(4th Euler), (Ara)-of-(2nd Zaniah), (2nd Conway)-of-(Gemini 110), (excentral)-of-(medial), (hexyl)-of-(Euler).

Note: A* is also the radical center of the incircle and the B- and C- excircles. (Nov 5, 2021)

A-vertex coordinates:

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

X:

X₃-ABC reflections

This triangle is obtained by reflecting the circumcenter X₃ about A, B, C. Reference: TCCT 6.13, p. 160.

Equivalences:

(anticomplementary)-of-(Johnson), (Johnson)-of-(anti-Ehrmann-mid).

Only for acute ABC: (hexyl)-of-(Ehrmann-side), (6th mixtilinear)-of-(2nd Euler).

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 the X-parabola-tangential triangle. (Reference: X12064)

Note: The X-parabola-tangential triangle is the anticevian triangle of X(115).

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_ACI_AB, I_BAI_BC, and I_CAI_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_BCI_CB, I_ACB'I_CA, I_ABI_BAC', and A'B'C' are congruent. The innermost triangle A'B'C' is called the Yff central triangle (Kimberling 1998, pp. 94-95). (See this figure in MathWorld).

A-vertex coordinates:

Trilinears 2*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 triangle is the cevian triangle of X(190).

Equivalences:

anticomplement of (Moses-Soddy).

A-vertex coordinates:

Trilinears 0 : -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 -- Yff Circles.). The circles in each triplet have radius r₁=r*R/(R+r) and r₂=r*R/(R-r), respectively. The centers of the circles in the first triplet are the vertices of the inner-Yff triangle. (Reference: Preamble just before X10037)

Equivalences:

(Johnson)-of-(1st Johnson-Yff).

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 -- Yff Circles.). The circles in each triplet have radius r₁=r*R/(R+r) and r₂=r*R/(R-r), respectively. The centers of the circles in the second triplet are the vertices of the outer-Yff triangle. (Reference: Preamble just before X10037)

Equivalences:

(Johnson)-of-(2nd Johnson-Yff).

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 the inner-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 the outer-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 triangle is 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. (MathWorld -- Yiu Triangle.)

A-vertex coordinates:

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

Yiu tangents

Let t_A be the tangent at A to the A-Yiu circle, and define t_B and t_C cyclically. Let A' = t_B∩t_C, and define B' and C' cyclically. Triangle A'B'C' is the Yiu tangents triangle. (Reference: X7495)

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. (MathWorld -- Yiu Triangle.)

A-vertex coordinates:

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

Z:

1st and 2nd Zaniah

In a triangle ABC, let A_mB_mC_m be the medial triangle, I the incenter and A' the touchpoint of the incircle with the side BC. Then the lines AA', IA_m and B_mC_m are concurrent at a point A₁. Denote B₁ and C₁ cyclically. The triangle A₁B₁C₁ is the 1st Zaniah triangle of ABC.

Continuing with the previous construction, let J_a be the A-excenter and A" the touchpoint of the A-excircle with the side BC. Then the lines AA", J_aA_m and B_mC_m are concurrent at a point A₂. Denote B₂ and C₂ cyclically. The triangle A₂B₂C₂ is the 2nd Zaniah triangle of ABC. (Reference: Preamble before X18214)

Equivalences:

1st Zaniah = complement of (extouch).

1st Zaniah = (Aries)-of-(2nd Zaniah).

2nd Zaniah = complement of (intouch).

2nd Zaniah = (anti-Ara)-of-(Wasat).

A-vertex coordinates:

1st Zaniah triangle:

Trilinears 2 : (a+b-c)/b : (a-b+c)/c
2nd Zaniah triangle:

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

Last changes:

2024

2024.06.09 - Added: 2nd Hatzipolakis-Moses / Published
2024.05.16 - Added: 1st and 2nd Pavlov
2024.04.25 - Added: 11th Brocard

2023

2023.12.20 - Added: more equivalences of triangles
2023.12.20 - Added: Aguilera, Aguilera-Pavlov
2023.10.10 - Added: K1343
2023.09.18 - Corrected: Mixtilinear triangles definitions
2023.09.04 - Corrected: 2nd Conway
2023.06.20 - Added: Moses-Miyamoto, Moses-Miyamoto-Apollonious
2023.01.15 - Added: Download coordinates link

2022

2022.12.28 - Added: Equivalences of triangles
2022.12.23 - Added: inverse-in-Conway
2022.12.15 - Added: BCE, BCE-incenters, BCI-excenters
2022.12.04 - Modified: Vijay-Paasche-tangents (same than Vijay-Paasche-polar), Vijay-Paasche-Hutson (without index)
2022.11.08 - Added: Garcia-Moses
2022.11.04 - Steiner symmetrized
2022.06.06 - Added: orthocentroidal-isogonic
2022.03.26 - Simplified/Constructions added: 3rd and 4th anti-tri-squares
2022.03.03 - Circumnormal and circumtangential symmetrized
2022.02.10 - Added: Malfatti-outer and Malfatti-touchpoints

2021

2021.12.28 - Added: 8th and 9th mixtilinear
2021.11.08 - Added: Dao
2021.10.25 - Added: construction of anti-inner-Garcia, anti-Kenmotu-centers, anti-Lucas-homothetic
2021.10.12 - Added: 10th Brocard
2021.10.09 - Added: Description of 1st Savin triangle, Vijay 1st to 7th
2021.09.25 - Modified: Roussel triangle reference link
2021.09.01 - Added: 1st & 2nd Kenmotu-centers
2021.08.21 - Added: Vijay-Paasche-Hutson 1 to 31, 1st- & 2nd- Savin, Moses-Soddy
2021.05.06 - Added: anti-1st Parry, anti-2nd Parry, anti-3rd tri-squares-central, anti-4th tri-squares-central, anti-X3-ABC reflections, anti-inner-Yff, anti-outer-Yff
2021.05.06 - Added: anti-1st Auriga, anti-2nd Auriga, anti-Ehrmann-mid, anti-inner-Garcia, anti-1st Kenmotu-free-vertices, anti-2nd Kenmotu-free-vertices
2021.04.24 - Added: note to 4th mixtilinear
2021.04.04 - Added: anti-1st circumperp-tangential
2021.02.24 - Page style changed.
2021.02.24 - Modified: Link in 7th Brocard triangle. Modified: Wolfram's MathWorld links.
2021.01.12 - Added: Constructions of half-diamonds triangles.
2021.01.12 - Added: 1st and 2nd Altintas-isodynamic triangles.
2021.01.12 - Modified: Link in 1st and 2nd half-diamonds triangles.

2020

2020.11.14 - Added: construction of anti-Ursa minor triangle.
2020.10.03 - Added: 7th-, 8th- and 9th- Brocard triangles.
2020.05.02 - Added: Vijay-Paasche-Hutson 1st/2nd/3rd, Vijay-Paasche-midpoints, Vijay-Paasche-polar, Paasche-Hutson
2020.04.13 - Modified: italic typesetting changed to bold in all triangles definitions. Triangles count corrected.
2020.04.12 - Added: Jenkins (1st, 2nd & 3rd), Jenkins-contact, Jenkins-tangential
2020.04.12 - Modified: X-parabola-tangential reference link
2020.02.25 - Added: largest-circumscribed-equilateral
2020.01.23 - Added: anticevian equivalences
2020.01.21 - Added: Vu-Dao-isodynamic equilateral triangles
2020.01.20 - Added: infinite altitude

2019

2019.12.24 - Added: Moses-Steiner osculating, Moses-Steiner reflection, orthic axes
2019.12.23 - Added: Bankoff equilateral, Bevan-antipodal, inverse-in-excircles, Yiu tangents, Kenmotu free vertices.
2019.10.08 - Added: Gemini triangles.
2019.07.21 - Added: 1st to 4th inner- and outer- Fermat-Dao-Nhi.
2019.07.05 - Added: 1st & 2nd half-diamonds, 1st & 2nd half-diamonds-central and 1st & 2nd half-squares.
2019.07.05 - Updated: URL of Bernard Gibert webpage.
2019.06.06 - Moved: AAOA
2019.05.17 - Added: 2nd and 3rd inner- and outer-Vecten.
2019.04.24 - Added: 3rd Hatzipolakis. Corrected: Hatzipolakis-Moses
2019.04.22 - Added: Walsmith
2019.03.15 - Added: 3rd and 4th isodynamic-Dao
2019.02.18 - Added: Soddy, 2nd inner-Soddy, 2nd outer-Soddy

2018

2018.12.19 - Corrected: excenters-midpoints
2018.12.10 - Moved: AAOA
2018.10.22 - Corrected: 7th & 8th Fermat-Dao
2018.10.01 - Corrected: 2nd Fuhrmann
2018.09.16 - Minor changes. Added triangle anti-Ursa-minor
2018.09.01 - Corrected: Lucas(-1) 2nd secondary tangents
2018.08.27 - Added: Wasat and anti-Wasat
2018.06.16 - Modified: K798e/K798i
2018.06.16 - Added: anti-Atik, 2nd anti-circumperp-tangential, anti-(inner/outer)-Grebe, anti-Honsberger, 1st anti-orthosymmedial, 1st anti-Sharygin, 3rd and 4th anti-tri-squares
2018.05.17 - Added: 2nd anti-extouch, Lucas antipodal tangents
2018.05.14 - Added: Notes to anti-triangles.
2018.05.04 - Added: Ehrmann -circumscribing, -inscribed, -cross, -mid, -side & -vertex.
2018.05.02 - Added: Zaniah triangles
2018.04.20 - Added: excosine triangles
2018.04.14 - Added: Ursa-major, Ursa-minor and Morley-midpoint triangles
2018.04.03 - Modified: Lucas triangles
2018.04.03 - Added: Morley-adjunct-midpoint triangles
2018.04.02 - Added: Fermat-Dao 11th to 16th, isodynamic-Dao 1st and 2nd, Lemoine-Dao 1st and 2nd
2018.03.22 - Added: Malfatti
2018.03.20 - Added: inner- and outer- Fermat and Fermat-Dao 1st to 10th
2018.02.12 - Added: Garcia-reflection, 2nd. Fuhrmann
2018.01.18 - Added: 1st and 2nd circumperp tangential, Hatzipolakis-Moses, 1st Brocard-reflected, K798.

2017

2017.12.20 - Added: excenters-reflections and anti-excenters-reflections
2017.11.06 - Added: altimedial and related
2017.09.10 - Modified: Atik
2017.09.03 - Added: Le Viet An
2017.07.10 - Added: tri-squares
2017.02.02 - Added: Hung-Feuerbach
2017.01.14 - Added: inner-Conway

2016

2016.12.28 - First published

INDEX OF TRIANGLES REFERENCED IN ETC

A:

ABC

AAOA (antiAOA)

ABC-X3 reflections

AaBbCc and (AaBbCc)*

Aguilera

Aguilera-Pavlov

AiBiCi and (AiBiCi)*

altimedial

altimedial orthic axes (AOA)

1st and 2nd Altintas-isodynamic

Andromeda

adjunct anti-altimedial

anticomplementary

anti-triangles

anti-altimedial

anti-Aquila

anti-Ara

anti-Ascella

anti-Atik

anti-1st/2nd Auriga

1st anti-Brocard

4th anti-Brocard

5th anti-Brocard

6th anti-Brocard

1st anti-circumperp

1st anti-circumperp-tangential

2nd anti-circumperp-tangential

(1st) anti-Conway

2nd anti-Conway

anti-Ehrmann-mid

anti-Euler

3rd anti-Euler

4th anti-Euler

anti-excenters-incenter reflections

anti-excenters-reflections

2nd anti-extouch

anti-inner-Garcia

anti-inner-Grebe

anti-outer-Grebe

anti-Honsberger

anti-Hutson intouch

anti-incircle-circles

anti-inverse-in-incircle

anti- 1st/2nd Kenmotu-centers

anti-1st/2nd Kenmotu-free-vertices

anti-Lucas(-1) homothetic and anti-Lucas(+1) homothetic

anti-Mandart-incircle

anti-McCay

6th anti-mixtilinear

anti-orthocentroidal

1st anti-orthosymmedial

anti-1st Parry

anti-2nd Parry

anti-reflection

1st anti-Sharygin

anti-tangential-midarc

3rd anti-tri-squares

4th anti-tri-squares

anti-3rd/4th tri-squares-central

anti-Ursa minor

anti-Wasat

anti-X3-ABC reflections

anti-inner/outer Yff

Antlia

AOA

Apollonius

Apus

Aquila

Ara

Aries

Artzt

Ascella

Atik

1st Auriga

2nd Auriga

Ayme

B:

Bankoff equilateral

ABC-X₃ reflections