## 13274Re: Concurrencies on a trianle.

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• Jun 13, 2006
--- In Hyacinthos@yahoogroups.com, "Kostas Vittas" <vittasko@...>
wrote:

> Dear all my friends.
>
> I would like to present an interesting configuration as
>follows : ( I apologize for my English )
>
> A triangle ABC is given and let AD, BE, CF be, it's altitudes.
> We draw three circles (K1), (K2), (K3), with side segments BC, AC,
>AB respectively, as diameters.
>
> We denote as D', D'', the intersection points of line AD from
> circle (K1). Similarly we denote the pair of points E', E'' and F',
> F'', on lines BE, CF, respectively. ( The points D', E', F',
>inwardly to the triangle ABC ).
>
> We denote A', as the intersection point of the lines E'F'',
> E''F'. Similarly B' as the one of the lines D'F'', D''F' and C' as
> the one of the lines D'E'', D''E'.
>
> Results :
>
> 1. The points A', B', C', lie on BC, AC, AB, respectively.
> 2. The lines AA', BB', CC', are concurrent at a point, so be it M.
> 3. The lines A'D',B'E',C'F', are concurrent at a point, so be it L.
> 4. The lines A'D'',B'E'',C'F'',are concurrent at a point,so be it
N.
> 5. The points L, M, N, lie on a line pass through orthocenter H,
> of the triangle ABC.
> 6. The points L, N, are the circumcenters of the triangles D'E'F',
> D''E''F'', respectively.
> 7. If D1, E1, F1, are the intersection points of circles (K1),
> (K2),(K3) respectively, from the circumcircle of the triangle
> D''E''F'', then the lines AD1, BE1, CF1, are concurrent.
> 8. If D2,E2,F2, are the intersection points of circles (K1),(K2),
> (K3)respectively, from the circumcircle of the triangle D'E'F',
> then the lines D1D2, E1E2, F1F2, are concurrent at orthocenter
> H of ABC.
>
> ...

Dear all my friends.

I would like to present the proof of proposition (7), as
follows: ( my drawing is erected on a triangle with sides
BC = 7.5, AC = 6.8, AB = 5.2 )

7a. Let A' be the intersection point of E1E'',F1F'' and B', the
intersection point of D1D'', F1F''and C',the intersection point
of D1D'', E1E''.

7b. The point A' is the radical center of the circles (K2),(K3),(M)
( the circumcircle of the triangle D''E''F'' ). That is the
segment line AD, passes through the point A'. Similarly the
segment lines BE, CF, pass through the points B',C'respectively.

7c. Let R be the intersection point of BC, D1D''. We draw a line
perpendicular to the segment line BC at point R, which cuts the
segment lines BD1, CD1, at points Q, S respectively.

7d. The point C, is the orthocenter of the triangle BQS ( because of
the lines BR,SD1,are perpendicular to the lines QS,BQ
respectively ).Hence the segment lines QC, BS,are perpendicular
each other and their intersection point, so be it D'1, lies on
the circle (K1).

7e. We will prove that the point D'1 coincides to the point D'.
Really from the cyclic quadrilateral CD1QR we have that
<RCQ = <RD1Q (1). But <RCQ = <BCD'1 (2) and <RD1Q = <BD1D''(3).
From (1),(2),(3) => <BCD'1 = <BD1D'' => BD''= BD'1 (4)
Hence the point D'1 coincides to the point D'. That is, we have
that the segment line CD' passes through the point Q.

7f. As a helping proposition, we use the result that the points B',
A, S, are collinear ( we shall prove the proposition for this
result, below ). Hence, we consider the segments AD', BB' CC',
concurrent at orthocenter H of ABC and then, by the Lemma 2, we
have that the points Q' ( the intersection point of AC', CD' ),
R, S, as the intersection points of the opposite sides, the
respectively, are collinear. Hence the point Q' coincides to
the point Q and then we have that the segment line AC', passes
through the point Q.

7g. Now, because of the points Q, R, S, are collinear, we conclude
that the triangles AB'C', BCD1, are perspective at one point, so
be it A1. That is the point A1, as the intersection point of
BC',B'C,lies on AD1. Similarly The point B1, as the intersection
point of AC',A'C and the point C1, as the intersection point of
AB', A'B, lie on BE1, CF1 respectively.

7h. We consider the segments AA',BB',CC', concurrent at orthocenter
H of ABC. So, by the Lemma 1, we have that the lines AA1, BB1,
CC1, are concurrent. Hence the lines AD1, BE1, CF1, are
concurrent at one point, so be it P and the proof of proposition
(7), is completed.

7.1 Helping Proposition. A triangle ABC is given. Let AD, BE,
CF be, it's altitudes and let H be, it's orthocenter. We draw
the circumcircle (K1) of the cyclic quadrilateral BDHF and we
denote as G, the intersection point of (K1), DE. Also we denote
as L, the intersection point of BE, FG. We consider an arbitrary
point between F, L and we denote as N, the intersection point of
(K1), HP.
If R, is the intersection point of BN, FG and Q, is the one
of BP, EG, prove that the points Q, R, A, are collinear.

7.1a Proof. We construct the line(e),through H and
perpendicular to the line BE. We denote as F', R', L', the
intersection points of HF, HR,(e)respectively, from the line BP.

7.1b It is easy to prove that the segment line FG,is
perpendicular to the BE ( from cyclic quadrilateral HDCE we
have that <EHC = <EDC, etc ). So, the point P, is the
orthocenter of the triangle RBH and then we have that the
segment line BR', is perpendicular to HR. Hence the point R',
lies on circle (K1) ( BH is diameter of (K1) ).

7.1c The points Q, R, A, lie on the side lines BP, FP, BF
respectively, of the triangle BFP. So, in order to these points
are collinear, it is enough to prove that is true, the
equality: [(RP):(RF)]·[(AF):(AB)]·[(QB):(QP)] = 1 (1)
( Menelaus theorem )

Because of the segment lines FL, AE are parallel,
=> (AF):(AB) = (EL): (EB) (2)

Also from the triangle BLP, by Menelaus theorem, we have that
=> [(EL):(EB)]·[(QB):(QP)]·[(GP):(GL)] = 1 (3)

By (2),(3),the equality (1) becomes:(RP): (RF)=(GP):(GL) (4)

From (4) => (RPRF):(RF)=(GPGL):(GL) => (FP):(RF)=(LP):(GL)

=> (FP):(RF) = (LP):(FL) (5) ( because of FL = GL )

Hence it is enough to prove that is true the equality (5),

7.1d We compare the bundles of lines B.LPFR and H.L'R'F'P.
Their homologous lines, form congruent angles ( <LBP = <L'HR',
<PBF = <R'HF', <FBR = <F'HP ).

So, their cross-ratios are congruent. That is we have
that ( L,P,F,R ) = ( L',R',F',P ) (6)

Because of ( L',R',F',P ) = ( P,F',R',L' )
=> ( L,P,F,R ) = ( P,F',R',L' ) (7)

Also because of PR //(e) we have that
( P,F',R',L' )=( P,F,R ) (8)

From (7),(8),=> ( L,P,F,R ) = ( P,F,R ) (9)

From (9) => [(FL):(FP)]:[(RL):(RP)] = (RP):(RF)

=> [(FL):(FP)]·[(RP):(RL)] = (RP):(RF)

=> (FL):(FP) = (RL):(RF) => (FP):(RF) = (FL):(RL) (10)

7.1e From (5),(10),=>(LP):(FL)=(FL):(RL)=>(FL)²=(LP)·(RL) (11)

From the orthogonal triangle BFH ( <BFH = 90º )
=>(FL)²=(LB)·(LH) (12)

From (11),(12), => (LP)·(RL) = (LB)·(LH) (13)

Hence it is enough to prove that is true the equality (13),

Really, this equality is true because of the similarity of the
orthogonal triangles, PLH and BLR ( <PLH = 90º, <BLR = 90º ).

We conclude now, that is also true the equality (1). Hence the
points Q, R, A, are collinear and the proof of proposition
(7.1), is completed.

Sometimes ( many times for me ), our mind follows a complicated
or a long way for the solution of a problem. So, I am waiting