Consider the good parametrization, which to each point E on the conic associates the point E' on BC, E' being the intersection of BC with DE. If E' has the coordinates E' = sB + tC, then E = D + kE' = D + k(sB + tC). Since D = A+B+C, this implies that E = (A+B+C) + k(sB + tC) = A + (1+ks)B + (1+kt)C. Since E is on the conic its coordinates must satify x

Substitution into the equation for E, gives E = D -(s+t)((1/t)B + (1/s)C) = A - (s/t)B - (t/s)C. Equivalently, by multiplying with st ==> E = stA - s

This shows that the inverse of the good parametrization considered is as stated (x

The proof for the general case of good parametrization (s',t') results from the fact that two such parametrizations are related through a linear relation (s'=as+bt, t'=cs+dt). This, in turn follows also by an argument similar to the previous one, by first looking at two good parametrisations for the same conic, defined as above. Should the parameters (s,t) and (s',t') represent the same point E in the two systems, then sB+tC = s'B'+t'C'. Writing B' = uA+vB+wC, C'=u'A+v'B+c'C and equating we get the desired relation.

If the good parametrization uses another line e different from AB, then the correspondence E'<-->E'' is described also through a linear relation between the projective coordinates of the two lines (see RectHypeRelation.html , where the relation is equivalently derived for the quotiens s/t, s'/t' of the projective coordinates).

Chasles_Steiner_Envelope.html

Complex_Cross_Ratio.html

Complex_Cross_Ratio2.html

CrossRatio.html

GoodParametrization.html

Harmonic_Bundle.html

HomographicRelation.html

HomographicRelationExample.html

ParabolaProperty.html

ProjectiveBase.html

ProjectiveCoordinates.html

ProjectivePlane.html

RectHypeRelation.html

Stereographic.html

Berger Marcel, Pansu Pierre, Berry Jean-Pic, Saint-Raymond Xavier

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