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Here is a comparative description of Morley triangles and cubics. This page was made with contributions by Edward Brisse.

Denote by T1, T2, T3 the three Morley triangles and by T1*, T2*, T3* the three corresponding Morley adjunct triangles (see TCCT, pp. 165-167, §§ 6.25, 6.26, 6.27, 6.30). IaIbIc is the excentral triangle.

Perspective triangles and perspectors :

 

T1

T2

T3

T1*

T2*

T3*

ABC is perspective at

X(357)

X(1136)

X(1134)

X(358)

X(1137)

X(1135)

 

 

perspector

T1

T1*

X(356), centroid of T1 and pivot of K029

T2

T2*

X(356)- = X(3276), centroid of T2 and pivot of K031

T3

T3*

X(356)+ = X(3277), centroid of T3 and pivot of K030

 

T1

T2

T3

T1*

T2*

T3*

IaIbIc is perspective at

X(1507)

X(6208)

X(6209)

X(1508)

X(6206)

X(6207)

Each point P related to T1 has a "plus" version P+ and a "minus" version P- obtained when the trisectors of A, B, C are replaced by those of A+2pi, B+2pi, C+2pi and A-2pi, B-2pi, C-2pi respectively. For example, X(1134) = X(357)+ and X(1136) = X(357)-.

Points on the cubics :

The Morley cubic contains A, B, C, X(1), Ia, Ib, Ic and

K029

K030

K031

the vertices of Ti and Ti*

T1 and T1*

T3 and T3*

T2 and T2*

its pivot

X(356)

X(3277) = X(356)+

X(3276) = X(356)-

the isogonal conjugate of its pivot

X(3605) = X(356)*

X(3607) = X(356)+*

X(3606) = X(356)-*

the perspector of ABC and Ti

X(357)

X(1134)

X(1136)

the perspector of ABC and Ti*

X(358)

X(1135)

X(1137)

the perspector of IaIbIc and Ti

X(1507)

X(1507)+

X(1507)-

its isogonal conjugate

X(1507)*

X(1507)+*

X(1507)-*

the perspector of IaIbIc and Ti*

X(1508)

X(1508)+

X(1508)-

its isogonal conjugate

X(1508)*

X(1508)+*

X(1508)-*

other centers

X(1134), X(1135)

X(1136), X(1137)

X(357), X(358)

other points

cevians of X(356)

cevians of X(356)+

cevians of X(356)-

Related trilinear coordinates :

The following table gathers together some points with first trilinear coordinate containing A/3 angles.

t

cos(A/3 + t)

sin(A/3 + t)

sec(A/3 + t)

csc(A/3 + t)

 

on the line X(16)-X(358)

on the circum-conic through X(14), X(357)

- π/3

X(1135)

X(3274)

X(1134)

X(3602)

- π/6

X(3273)

X(1137)

X(3604)

X(1136)

0

X(358)

X(3275)

X(357)

X(3603)

π/6

X(3274)

X(1135)

X(3602)

X(1134)

π/3

X(1137)

X(3273)

X(1136)

X(3604)

 

 

 

 

 

Remark : the line X(16)-X(358) also contains all the points with first trilinear coordinate

• cos A + 2 cos B/3 cos C/3 + t cos (A/3 + u),

• cos A + 2 sin B/3 sin C/3 + t cos (A/3 + u),

where t and u are any real numbers.
table9a

Other related curves :

See the pages K587, Q002, Q003, Q042, Q043.