Trisectors and perspective,
an entry to complexity in geometry

Bruno Autin, CERN Honorary member

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Abstract

The symbolic calculation of Wolfram Language  is applied to the topology of triangles created by intersecting the trisectors of a reference triangle T. A special set of intersections  is the equilateral Morley triangle M. In addition, some lines joining the vertices of T and M intersect at the same point called the second Morley center X(356) in Kimberling’s notation. T and M are thus in a perspective characterized by a center and a Desargues line. In fact, all the triangles of the set of intersections are in perspective with T and between them. That tessalation of a triangle by trisectors thus generates a family of pespectives to which is associated a space of perspectives whose elements are the centers and Desargues line of each perspective. Desargues lines can be arranged in pencils or in triangles.

Linear perspective

Definition and Desargues theorem

Two triangles ABC and DEF are in perspective when the lines AD, BE and CF, called lines of perspective, intersect at the same point M, the center of perspective. The example applies to the triangle of the pedals of the altitudes of an integer triangle T whose side lengths are 13, 14, 15. The center of perspective is the orthocenter of T and the lines of sight are AD, BE, CF. 
As a consequence of the definition, the intersections of the sides in perspective : BC⋂EF, CA⋂FD, AB⋂DE are aligned. This is Desargues’theorem which is at the origin of projective geometry. The elements of a perspective are given by the Geometrica function Desargues.

t=ETriangle[];
al=Altitudes@t;
si=Sides@t;
pe=Intersections[al,RotateLeft@si];
ta=ETriangle@@pe;
Desargues[t,ta]

Figure 1.	Perspective of the reference triangle with the triangle of the pedals of the altitudes.

The perspective can be considered as a relation 𝒫 in the set of triangles whose properties are examined in the next sections

Properties

Reflexiveness

The perspective is reflexive (T 𝒫 T), a triangle is in perspective with itself. The center of perspective is then any point of the plane, the sides are their own images and the Desargues line lies at infinity.

Get["/Users/brunoautin/Geometrica.m"]

Desargues[ETriangle[CPoint[0,0],CPoint[1,0],CPoint[1/3,1]],ETriangle[CPoint[1,0],CPoint[0,0],CPoint[1/3,1]]]

Figure 2.	Perspective of the reference triangle with itself.

Symmetry

The perspective is symmetric

(Subscript[T, 1])𝒫 (Subscript[T, 2])= (Subscript[T, 2])𝒫 (Subscript[T, 1])

Transitivity

The perspective is not transitive in general. The triangles of the pedals of the altitudes and of the pedals of the bisectors are both in perspective with the reference triangle.

t=ETriangle[];
bi=Bisector[t,Angle->True];
si=Sides@t;
pe=Intersections[bi,RotateLeft@si];
tb=ETriangle@@pe;
Desargues[t,tb]

Figure 3.	Perspective of the reference triangle with the triangle of the pedals of the bisectors.

However they are not in perspective between them.

Desargues[ta,tb]

Perspective is transitive for triangles built on the same pencil of three lines.

t=ETriangle[];
{tp1,tp2}=Simplify/@Homothety[t,t[[1]],{2,Pi}];
FullSimplify@Desargues[t,tp1]
FullSimplify@Desargues[t,tp2]
FullSimplify@Desargues[tp1,tp2]

Figure 4.	Perspective of homothetic triangles.

Uniqueness

Usually, when two triangles are in perspective, the perspective is defined by one center of symmetry and one Desargues line. However, the previous example shows perspectives with two centers as it is the case for all triangles with a common vertex. Moreover, several triangles, in fact an infinity, may share the same perspective. This is easily understood for a 3D perspective defined by a center c and a triangle t outside of c, all the sections of the pyramid (c,t) by a pencil of planes belong to the perspective defined by c and the axis of the pencil, its Desargues line.

Automatic perspectives

Two triangles which have one or more vertices in common are automatically in perspective

Desargues[ETriangle[CPoint[0,0],CPoint[1,0],CPoint[1/3,1]],ETriangle[CPoint[1,1/2],CPoint[0,0],CPoint[2,1/3]]]

Figure 5.	Automatic perspective of two triangles with one vertex in common.

The triangles are drawn in bold black. The center and lines of sight of one perspective are in red for one and in blue for the other. The common Desargues line is in magenta, it joins the common vertex and the intersection of the lines collinear with the sides opposite to the common vertex.

Elements at infinity

As seen in the definition of a perspective, the center and Desargues line involve intersections of lines. In order to avoid the special case of parallel lines, the projective point is introduced. It has three components and is related to the cartesian point by the relation

ProjectivePoint[x,y,t]= CPoint[(x/t),(y/t)]

t is a constant of proportionality usually set to 1. However, and this is the interest of the concept, t may be zero when a point results from the “intersection” of two parallel lines

ProjectivePoint[-v,u,0]= CLine[u,v,w]⋂ CLine[k u, k v, l w]

The first two components are those of a vector (-v,u) collinear with the lines since (u,v) is a normal vector. Again those components are defined within a constant of proportionality:

ProjectivePoint[-v,u,0]= ProjectivePoint[-(v/(Sqrt[(u^2)+(v^2)])),(u/(Sqrt[(u^2)+(v^2)])),0]

The components are then those of a unit vector. The third component when equal to 0 means that the point lies on the InfinityLine

t=0

Family of perspectives

Complexity appears when several triangles are in perspective. It is precisely what happens with the tessellation of a reference triangle T into sub-triangles produced by the intersections of the trisectors of T. The trisectors divide each angle of T into three equal parts in the same way as bisectors divide angles in two equal parts. There are inner and outer trisectors but the study is limited to inner trisectors, outer trisectors having similar properties. In fact, the tessellation of a triangle with arbitrary pencils of four lines centered at the vertices may produce also a family of perspectives but such a configuration has not been tested in this paper. The study presented here is actually a generalisation of Morley theorem [1] which states the existence of an equilateral triangle in the set of the intersections of the trisectors. Moreover, it has been proven later that the Morley triangle is in perspective with T [2].

The six trisectors are given y the function NSectors

trs=Flatten@NSectors[t,3]

They generate 15 intersections,

itr=Map[FullSimplify,Map[TrigReduce,Flatten@Table[Intersections[trs[[i]],trs[[j]]],{i,2,6},{j,1,i-1}],{1}],{1}];

B, C and A correspond to (1,2), (3,4) and (5,6). There are thus 12 inner intersections. The triangle (3, 6, 11) is the equilateral Morley triangle.

{a,b,c}=Vertices@ETriangle[];
itr0=Fold[DeleteCases,itr,{a,b,c}];(*inner intersections of the trisectors*)
tMorley=ETriangle[itr0[[3]],itr0[[11]],itr0[[6]]];
Draw[DrawRange[-9,9],LineOrigin@CenterOfGravity[t],t,Paint[tMorley,Gray],Green,trs,Blue,Legend[ToString/@Range[12],itr0],Red,Legend[ToString/@Range[6],trs]]

Figure 6.	Trisectors and their intersections inside the triangle of reference. The Morley triangle is painted in gray.

One may distinguish three subsets of triangles, those which have two, one or no vertices in common with T. The first two subsets are known to be automatically in perspective with T. For the third subset, the property is not obvious but turns out to be true and would deserve a general proof not given here. What this paper proposes is an organisation of the perspectives in a general perspective space whose elements are the centers which may be regrouped in a special configuration and the Desargues lines which form either pencils or triangles called Desargues triangles.

Inner triangles with a side of the reference triangle in common

Twelve triangles have the side BC in common with T. They form the family (Subscript[T, BC]).

Figure 7.	(Subscript[T, BC])family.

In total, (Subscript[T, BC]), (Subscript[T, CA]) and (Subscript[T, AB]) have thus 36 perspectives with T. In fact, it is sufficient to study the family (Subscript[T, BC]) since (Subscript[T, CA]) and (Subscript[T, AB]) have similar characteristics. The image of their perspectives with T is shown in Figure 6. All the centers of perspective are distinct and located on BC and the Desargues line is always collinear with BC.

tbc=ETriangle[b,c,itr0];
ptbc=Desargues[N@t,N@tbc];
tptbc=Transpose@ptbc;
Show[tptbc[[1]],Draw[Blue,Legend[ToString/@Range[12],itr0]]]

Figure 8.	(Subscript[T, BC])perspectives with reference triangle.

The cross perspectives of the triangles of the family (Subscript[T, BC]) belong to a Pascal triangle of length 11 and thus amount to 55.

ptbc1=Table[Desargues[N@tbc[[i]],N@tbc[[j]]],{i,2,12},{j,1,i-1}];
tptbc1=Transpose/@ptbc1;
Show[Draw[t],Flatten@Table[tptbc1[[i,1]],{i,11}],Draw[Blue,Legend[ToString/@Range[12],itr0]],PlotRange->{{-10,15},{-1,15}}]

Figure 9.	Cross perspectives in (Subscript[T, BC]).

They are represented in Figure 9. The perspective space contains a single Desargues line, the one collinear with BC. The centers of perspective are not all distinct. The perspectives between triangles whose vertices lie on a same trisector are the same. B, C and the intersections of the trisectors issued from A with BC accumulate four centers of perspective each. In addition, the couples of perspectives ((11, 5), (10, 8)) and ((12, 6), (9, 7)) are the same at least within numerical precision. The first common center has the abscissa -5.47593 and the second one 2.17611. It may be noted that 6 and 11 are vertices of the Morley triangle drawn in gray in Figure 8. The Desargues lines are always collinear with BC. The triangles of base CA and AB produce the same type of perspective and the Desargues triangle of all the perspectives is T. In total, there are 165 perspective.

Figure 10. Special cross perspectives in (Subscript[T, BC]) that share same center and Desargues line..

Inner triangles with a vertex of the reference triangle in common

Perspectives with the reference triangle

The method followed for triangles with a common side can be resumed for triangles with a common vertex. (Subscript[T, A]), (Subscript[T, B]) and (Subscript[T, C]) are the families of perspectives with T of the triangles built on the intersections of the trisectors of T. Moreover, there are 66 triangles per vertex since the free vertices belong to the Pascal triangle of the 12 inner intersections.

tA=N@Flatten@Table[ETriangle[a,itr0[[i]],itr0[[j]]],{i,2,12},{j,1,i-1}];
dtA=Draw[DrawRange[-9,9],LineOrigin@CenterOfGravity[t],t,Paint[tMorley,Gray],Green,trs,Blue,tA];
tB=N@Flatten@Table[ETriangle[b,itr0[[i]],itr0[[j]]],{i,2,12},{j,1,i-1}];
dtB=Draw[DrawRange[-9,9],LineOrigin@CenterOfGravity[t],t,Paint[tMorley,Gray],Green,trs,Blue,tB];
tC=N@Flatten@Table[ETriangle[c,itr0[[i]],itr0[[j]]],{i,2,12},{j,1,i-1}];
dtC=Draw[DrawRange[-9,9],LineOrigin@CenterOfGravity[t],t,Paint[tMorley,Gray],Green,trs,Blue,tC];
sb=SubscriptBox["T",#]&/@{"A","B","C"}//DisplayForm;
Grid[{{(Subscript[T, A]),(Subscript[T, B]),(Subscript[T, C])},{dtA,dtB,dtC}},Frame->All]

Figure 11.	(Subscript[T, A]) , (Subscript[T, B]) and (Subscript[T, C]) families with their 198 triangles.

As already mentioned, the perspectives are double for triangles with one common vertex. The set of perspectives is given by the function:

{ptA,ptB,ptC}=Desargues[N@t,N@{tA,tB,tC}];

it is convenient to use the transpose of ptA, ptB, ptC whose second and third elements are the centers of perspective and Desargues lines.

{tptA,tptB,tptC}=Transpose/@{ptA,ptB,ptC};

Centers of perspective

Symbolic computations become exceedingly long, at least for a personal computer. The arguments of the functions are thus first converted into real numbers then rationalized within (10^(-11)) precision.

{ceA,ceB,ceC}=Flatten/@{tptA[[2]],tptB[[2]],tptC[[2]]};
{ceAr,ceBr,ceCr}=Rationalize[#,Power[10,-11]]&/@{ceA,ceB,ceC};
Length/@{ceAr,ceBr,ceCr}

There are 132 centers per perspective as expected since each perspective has two centers. However, some centers may be the same and the distinct centers are given by the union of the 396 centers

{uceAr,uceBr,uceCr}=Union/@{ceAr,ceBr,ceCr};
Length/@{uceAr,uceBr,uceCr}

The common points to the 108 centers are given by the intersection of the three sets

ice=Intersection[uceAr,uceBr,uceCr];
dice=Draw[DrawRange[-8,9],LineOrigin@CenterOfGravity[t],Paint[{t,tMorley},{Gray,Blue}],Green,trs,AbsolutePointSize[2],Orange,Bisector[t,Angle->True],Yellow,ice]

Figure 12.	The 20 centers common to (Subscript[T, A]) , (Subscript[T, B]) and (Subscript[T, C]) families.

There are 16 centers proper to each family, they are given by the complement of the set of their centers to the intersection of the families.

{cuceAr,cuceBr,cuceCr}=Complement[#,ice]&/@{uceAr,uceBr,uceCr};
dcce=Draw[DrawRange[-8,9],LineOrigin@CenterOfGravity[t],Paint[{t,tMorley},{Gray,Blue}],Green,trs,AbsolutePointSize[2],Orange,Bisector[t,Angle->True],Yellow,ice,White,cuceAr,LightBlue,cuceBr,Cyan,cuceCr]

Figure 13.	The 68 centers of the (Subscript[T, A]) (white), (Subscript[T, B]) (light blue) and (Subscript[T, C]) (cyan) families.

In Figure 13, one can note that clusters of 3 centers are distributed along the bisectors and that 9 centers gather near the incenter. A number of centers coincide with the intersections of the trisectors.

Desargues lines

The distribution of Desargues lines in the perspective space follows the same approach using the third element of the transposed matrix provided by the desargues function.

{deA,deB,deC}=Flatten/@{tptA[[3]],tptB[[3]],tptC[[3]]};
{deAr,deBr,deCr}=Rationalize[#,Power[10,-11]]&/@{deA,deB,deC};
Length/@{deAr,deBr,deCr}

There are as many Desargues lines as perspectives. As for the centers, one expects a number of common Desargues lines.

{udeAr,udeBr,udeCr}=Union/@{deAr,deBr,deCr};
Length/@{udeAr,udeBr,udeCr}

There is an uncertainty here on the number of distinct Desargues lines due to the simplification procedure adopted for their expression. In fact, the pencil of Desargues lines of each family contains the lines collinear to the sides issued from the vertex and, for some reason, the line BC is counted twice in the second pencil as shown in following tests

{l1,l2,l3}=ELine[{a,b,c},{b,c,a}];
Cases[#,True]&/@{IdenticalQ[l1,udeAr],IdenticalQ[l2,udeAr],IdenticalQ[l3,udeAr]}
Cases[#,True]&/@{IdenticalQ[l1,udeBr],IdenticalQ[l2,udeBr],IdenticalQ[l3,udeBr]}
Cases[#,True]&/@{IdenticalQ[l1,udeCr],IdenticalQ[l2,udeCr],IdenticalQ[l3,udeCr]}

The number of distinct lines in each pencil of Desargues lines is thus 32.

dude=Draw[DrawRange[-22,22],LineOrigin@CenterOfGravity[t],Paint[t,Yellow],Red,udeAr,Green,udeBr,Blue,udeCr]

Figure 14.	Perspective space of Desargues lines in (Subscript[T, A])(red), (Subscript[T, B])(green) and (Subscript[T, C]) (blue) family.

6 special triangles

The above figure shows that Desargues lines outline Desargues triangles in the perspective space. One may distinguish a special Desargues triangle whose vertices are the common intersections of red, green and blue Desargues lines. That configuration is produced by the subset of six triangles which have no trisectors in common, namely
((1, 2), (3, 5), (4, 6)), ((1, 2), (3, 6), (4, 5)), ((3, 4), (1, 5), (2, 6)), ((3, 4), (1, 6), (2, 5)), ((5, 6), (1, 3), (2, 4)), ((5, 6), (1, 4), (2, 3)) 
Using Figure 6, one may reconstruct a new family of triangles (Subscript[T, ABC]).

tABC=ETriangle[{b,b,c,c,a,a},{itr0[[7]],itr0[[11]],itr0[[5]],itr0[[9]],itr0[[1]],itr0[[3]]},{itr0[[12]],itr0[[8]],itr0[[10]],itr0[[6]],itr0[[4]],itr0[[2]]}];
tABC1=ETriangle[{b,c,a},{itr0[[7]],itr0[[5]],itr0[[1]]},{itr0[[12]],itr0[[10]],itr0[[4]]}];
tABC2=ETriangle[{b,c,a},{itr0[[11]],itr0[[9]],itr0[[3]]},{itr0[[8]],itr0[[6]],itr0[[2]]}];
Draw[t,Red,tABC1,Blue,tABC2]

Figure 14.	(Subscript[T, ABC])family made of 6 triangles which have no trisectors in common.

The perspectives with T are given by

ptABC=Desargues[N@t,N@tABC];
tptABC=Transpose@ptABC;

The 12 centers displayed in Figure 15 appear to coincide with the intersections of the trisectors.

cptABC=tptABC[[2]];
dcptABC=Draw[DrawRange[-8,9],LineOrigin@CenterOfGravity[t],Paint[{t,tMorley},{Gray,Blue}],Green,trs,AbsolutePointSize[2],Orange,Bisector[t,Angle->True],Yellow,cptABC]

Figure 15.	Centers of perspective with T of (Subscript[T, ABC]) family.

The procedure is repeated for the Desargues lines and confirms the property suggested by Figure 13.

dptABC=tptABC[[3]];
l1={dptABC[[1]],dptABC[[3]],dptABC[[5]]};
{u,v,w}=Intersections[l1,RotateLeft@l1];
tdABC=ETriangle[u,v,w];
ddptABC=Draw[DrawRange[-19,10],LineOrigin[CenterOfGravity@t],Paint[{t,tMorley},{Gray,White}],Red,tdABC,Blue,dptABC[[2]],dptABC[[4]],dptABC[[6]],Black,Legend[{"ABC","UVW"},{t,tdABC}]]

Figure 16. Perspective space of Desargues lines in (Subscript[T, ABC]).

Figure 16 represents the Desargues lines of perspective space of ((Subscript[T, ABC]) 𝒫 T). At each vertex of T, there are two Desargues lines: the red one corresponds to a regular triangle drawn in red in Figure 14 and the blue one defined by the perspective of a stretched blue triangle in Figure 14. The three red Desargues lines can be converted into the red triangle of Figure 16 whereas the blue lines form a pencil as shown by the next test.

ldde=(Flatten@{dptABC[[2]],dptABC[[4]],dptABC[[6]]})/.CLine->List;
Chop@Det[ldde]

They also contain the vertices of the Desargues triangle which is thus in perspective with T.

lvM=(Flatten@(ProjectivePoint/@(N@{u,v,w})))/.ProjectivePoint->List;
Chop@Table[Dot[lvM[[i]],#]&/@ldde,{i,3}]

In spite of the appearance, the vertices of the Morley triangle do not belong to the blue pencil.

Inner triangles disconnected from the reference triangle

All the triangles built on the inner intersections are formally produced by the list

ltrc=Flatten@Table[tr[i,j,k],{i,10},{j,i+1,11},{k,j+1,12}];

However, some sets of three points are located on the same trisector and create degenerate triangles.

ltrc0=tr@@@{{1,3,5},{1,3,9},{3,5,9},{9,10,11},{9,10,12},{10,11,12},{1,2,7},{1,2,11},{2,7,11},{2,4,6},{2,4,10}, {4,6,10},{5,6,7},{5,6,8}, {6,7,8},{3,4,8}, {3,4,12},{4,8,12}};

The genuine triangles are constructed with the tr function. They are contained in the list of 202 elements

Clear@tr
tr[m_Integer,n_Integer,p_Integer]:=ETriangle[itr0[[m]],itr0[[n]],itr0[[p]]];
ltrc1=N@Complement[ltrc,ltrc0];
Length@ltrc1

and the family of inner triangles (Subscript[T, I]) can be drawn

Draw[t,Blue,ltrc1]

Figure 17. The 202 triangles of the (Subscript[T, I]) family.

Perspectives with the reference triangle

Not all the inner triangles are in perspective with T.

des=Desargues[N@t,N@ltrc1];

In fact, 102 inner triangles are not in perspective with T.

nopt=Cases[des,Null];
Length@nopt

It thus remains 100 perspectives.

des1=Complement[des,nopt];
tdes1=Transpose@des1;
fdes1=Grid[Partition[tdes1[[1]],20],Frame->All,ItemSize->3.5]

Figure 18. The 100 perspectives of (Subscript[T, I]) family.

Out of the 100 above perspectives, the perspective with T can be proven by straightforward considerations for most but of them.

3-center perspectives

92 perspectives have a single center but the last eight have three centers which coincide with the vertices of the triangle.

Grid[{Table[tdes1[[1,i]],{i,93,100}]},Frame->All]

Figure 19. Three center perspectives of (Subscript[T, I]) family.

The Desargues line is always collinear with the side BC of the reference triangle.

Simplify@CLine[0,8.677925458231893`,0]

Simplify/@Table[tdes1[[3,i]],{i,93,100}]

Eight special perspectives

As for the perspectives of triangles having a vertex in common with T, one may examine the properties of the perspectives that involve triangles which have no trisectors in common:
((1,3), (2,5), (4,6)), ((1,3), (2,6), (4,5)), ((1,4), (2,5), (3,6)), ((1,4), (2,6), (3,5)), ((1,5), (2,3), (4,6)), ((1,5), (2,4), (3,6)), ((1,6), (2,3), (4,5)), ((1,6), (2,4), (3,5)).
That special family is called (Subscript[T, M]) because it contains the Morley triangle.

x1=Intersections[{trs[[1]],trs[[2]],trs[[4]]},{trs[[3]],trs[[5]],trs[[6]]}];
x2=Intersections[{trs[[1]],trs[[2]],trs[[4]]},{trs[[3]],trs[[6]],trs[[5]]}];
x3=Intersections[{trs[[1]],trs[[2]],trs[[3]]},{trs[[4]],trs[[5]],trs[[6]]}];
x4=Intersections[{trs[[1]],trs[[2]],trs[[3]]},{trs[[4]],trs[[6]],trs[[5]]}];
x5=Intersections[{trs[[1]],trs[[2]],trs[[4]]},{trs[[5]],trs[[3]],trs[[6]]}];
x6=Intersections[{trs[[1]],trs[[2]],trs[[3]]},{trs[[5]],trs[[4]],trs[[6]]}];
x7=Intersections[{trs[[1]],trs[[2]],trs[[4]]},{trs[[6]],trs[[3]],trs[[5]]}];
x8=Intersections[{trs[[1]],trs[[2]],trs[[3]]},{trs[[6]],trs[[4]],trs[[5]]}];
et=ETriangle@@@{x1,x2,x3,x4,x5,x6,x7,x8};
Draw[t,Blue,et]

Figure 20. Triangles of (Subscript[T, M]) family.

det=Desargues[N@t,et];
tdet=Transpose@det;
Grid[{tdet[[1]]},Frame->All]

Figure 21. The eight perspectives of (Subscript[T, M]) family.

By regrouping the centers in the perspective space, one notices that they are located on two pencils centered at 3 and 7

ctdet=tdet[[2]];
{da7,db7,dc7}=ELine[{a,b,c},ctdet[[7,1]]];
OfQ[{ctdet[[5]],ctdet[[2]],ctdet[[8]]},{da7,db7,dc7}]

{da3,db3,dc3}=ELine[{a,b,c},ctdet[[3,1]]];
OfQ[{ctdet[[4]],ctdet[[6]],ctdet[[1]]},{da3,db3,dc3}]

Draw[DrawRange[-9,9],LineOrigin@CenterOfGravity[t],Paint[t,Gray],Green,trs,AbsolutePointSize[2],Blue,da7,db7,dc7,Red,da3,db3,dc3,Yellow,ctdet,Legend[ToString/@Range[8],ctdet],Red,et[[3]],Blue,et[[7]]]

Figure 22. The (3, 7) double star of (Subscript[T, M]) family.

The two triangles of centers 3 (red) and 7 (blue) are drawn in the above figure. Whether they are equilateral or not is tested using the function EquilateralQ.

EquilateralQ/@{et[[3]],et[[7]]}

The red triangle is the Morley triangle. The other triangle has no special shape.

Desargues lines

There are only four distinct Desargues lines which can be collected in four triangles NQS, QRM, RPS, PNM. Those triangles are in perspective with T and give rise to a new level of perspectives. One thus observes the beginning of a process which may be endless.

Figure 23.	Desargues triangles of the perspectives of of (Subscript[T, M]) family.

Conclusion

A general study of the perspectives between a reference triangle T and triangles whose vertices are the intersections of the trisectors of T has been presented. They have been divided into three classes : triangles having a common side, a common vertex and no element in common. The definition of a perspective space has been introduced to regroup the elements characteristic of the perspectives namely centers and Desargues lines. With that description, some remarkable configurations appeared which situate the Morley theorem in a global context. The formation of Desargues triangles by intersection of Desargues lines in the perspective space opens a new realm of perspectives which has not been thoroughly investigated.