Resumen cuadro 6
Anexo 9.1 Realización del plan de clases
A line and a point are called incidental if the point belongs to the line or the line passes through the point.
Definition 40
A duality is a transformation which associates bi-univoc to a point a line. It is admitted that this correspondence preserves the incidental notion; in this mode to collinear points correspond concurrent lines and reciprocal.
If it I considered a theorem T whose hypothesis implicitly or explicitly appear points, lines, incident and it is supposed that its proof is completed, then if we change the roles of the points with the lines reversing the incidence, it is obtained theorem T’ whose proof is not necessary.
Theorem 40 (The dual of Pappus’ theorem)
If we consider two bundles each of three concurrent lines S a b c( , , ), '( ', ', ')S a b c such that the lines a b, ' and b a, ' intersect in the points C C1, 2 ; a c, ' and c a, ' intersect in the points
1, 2
B B and the lines b c, ' and c b, ' intersect in A A1, 2 , then the lines A A B B C C1 2, 1 2, 1 2 are
concurrent.
Proof
Analyzing the figure 59 we observe that it is obtain by applying the duality principle to Pappus’ theorem.
Indeed, the two bundles S a b c( , , ), '( ', ', ')S a b c correspond to the two triplets of vertexes of a hexagon situated on the lines d d1, 2 to which correspond the points S and 'S .
The Pappus’ theorem proves the collinearity of U V W, , which correspond to the
concurrent lines A A B B C C1 2, 1 2, 1 2. Observation 28
95 c’ a B1 b’ a’ C1 A1 b C2 S B2 A2 c S’ I Fig. 59 Theorem 41
We consider a complete quadrilateral and through the vertexes E F, we construct two
secants, which intersect AD BC, in the points E E1, 2 and AB CD, in the points F F1, 2. F A F1 B E1 E2 E C F2 D Fig. 60 I
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Then the lines E F F E1 2, 1 2 intersect on the diagonal AC and the lines E E F F1 2, 1 2 intersect on the
diagonal BD.
Indeed, this theorem is a particular case of the precedent theorem. It is sufficient to consider the bundles of vertexes E F, and of lines
(
CD E E AB, 1 2,)
respectively(
AD F F BC, 1 2,)
see figure 60, and to apply theorem 28.
From what we proved so far, it result that the triangles BF E1 2 and DF E2 1 are
homological, therefore BD F F E E, 1 2, 2 1 are concurrent. Remark 44
The dual of Pappus’ theorem leads us to another proof for theorem 34 (Rosanes). We prove therefore that two homological triangles are tri-homological.
We consider the triangles ABC A B C, ' ' ' bi-homological. Let S and S' the homology
centers: S the intersection of the lines AA BB CC', ', ' and
{ }
S' = AB'BC'CA'( see figure 61). We’ll apply theorem 41 for the bundles S AA BB CC(
', ', ')
and S C B A C B A'(
' , ' , ')
.A A1 C B S’ C’ B’ S Fig. 61 We observe that
{ }
' ' ' AAA C= A ,{ }
' ' ' C BBB = C ,{ }
' ' ' BBB A= B ,{ }
' ' CCA C= C ,97
{ }
' ' AAB A= A ,{ }
' ' ' CCC B= CTherefore the lines BA CB AC', ', ' are concurrent which shows that the triangles ABC and
' ' '
C A B are homological, thus the triangles ABC and A B C' ' ' are tri-homological.
Theorem 42
In triangle ABC let’s consider the Cevians AA BB CC1, 1, 1 in M1 and AA BB CC2, 2, 2
concurrent in the point M2. We note A B C3, ,3 3 the intersection points of the lines
(
CC BB1, 2)
,(
AA CC1, 2)
respectively(
BB AA1, 2)
, and A B C4, 4, 4 the intersection points of the lines(
CC BB2, 1)
,(
AA CC2, 1)
respectively(
BB AA2, 1)
, then(i) The triangles A B C3 3 3 and A B C4 4 4 are homological, and we note their homological center with P.
(ii) The triangles ABC and A B C3 3 3 are homological, their homological center being
noted Q.
(iii) The triangles ABC and A B C4 4 4 are homological, their homological center being noted with R
(iv) The points P Q R, , are collinear. Proof (i) A B1 C3 C2 B3 A4 B2 C1 M1 P A3 C4 B4 U A2 C B A1 Fig. 62
Let consider the point P the intersection of A A3 4 and C C3 4 with the sides of the
hexagon C M A A M C4 1 3 4 2 3, which has each three vertexes on the lines BB1, BB2. In conformity
with Pappus’ theorem the opposite lines C M4 1, A M4 2; M A1 3, M C2 3 ; A3A4 , C3C4 intersect in
collinear points.
These points are B B3, 4 and P; therefore the line B B3 4 passes through P, and thus the
triangles A B C3 3 3, A B C4 4 4 are homological. We note U V W, , their homological axis, therefore
{ }
U =B C3 3B C4 498
{ }
V =A C3 3A C4 4{ }
W =A B3 3A B4 4(ii)
We consider the hexagon C B M C B M4 4 1 3 3 2 , which each of its vertexes on AA1
respectively AA2.
The opposite sides
(
B C B C3 3, 4 4)
,(
C M C M3 1, 4 2)
,(
M B M B1 4, 2 3)
intersect in the collinearpoints U B C, , . It results that the point U is on the side BC and similarly the points V W, are on
the sides AC AB, .
Consequently the triangle ABC is homological with A B C3 3 3. (iii)
From the fact that U V W, , are respectively on BC AC AB, , , from their collinearity and from the fact that U is on B C4 4, V belongs to line A C4 4, and W belongs to the line A B4 4, it results that the triangles ABC and A B C4 4 4 are homological.
(iv)
The lines BC B C B C, 3 3, 4 4 have Uas common point, we deduct that the triangles BB B3 4 and CC C3 4 are homological. Consequently their opposite sides intersect in three collinear points, and these points are P Q R, , .
Remark 45
The point (iv) of the precedent theorem could be proved also by applying theorem 18. Indeed, the triangles
(
ABC A B C A B C, 3 3 3, 4 4 4)
constitute a homological triplet, and two bytwo have the same homological axis, the line U V W, , . It results that their homological centers
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Chapter 4
Homological triangles inscribed in circle
This chapter contains important theorems regarding circles, and certain connexions between them and homological triangles.