respectively {A,B,F}. Now vary a point P on (c) and draw lines AP, BP and their intersection points D, C on a, b

respectively. When F coincides with the middle of AB, lines f are parallel to e. When F does not coincide with the middle

of AB, all lines (f) pass through a point E on (e), which is the

cases are discussed respectively in Thales.html and Harmonic.html .

quadrangle ABCD has vertices {A,D} on line (a), vertices {B,C} on line (b) and intersection P of diagonals (AC,BD)

lying on line (c), then the intersection point E of its the side-pair (AB,CD) is always on a fixed line (red) which is the

harmonic conjugate of (c) with respect to (a,b).

e (alternatively passing through E), define the correspondence of intersection points D --> C. Introducing

line-coordinates on (a), (b) with origin at O, the first correspondence is described by a function of the form f(x) = ux, u

being a constant. In the second case the correspondence takes the form f(x) = px/(qx+r), with constants {p,q,r}, which

is a homographic relation. This is a functional view of Thales theorem. The second case is a generalization of this view.

The linear relation generalizes to a homographic relation between the two lines (a), (b), see RectHypeRelation.html for a

discussion on that. The subsequent remarks are comments on this functional view of Thales theorem and its

generalization.

(a) and (b). Again we construct lines (f) in the same way: taking arbitrary P on (c), drawing PA, PB and their

intercepts C, D with b, a respectively. Line (f) now envelopes a conic. For varying P on (c), the relation between line

coordinates on a, b is again a homographic one: y = (px+q)/(rx+s) with non-zero q.

The conic is tangent to lines {a, b, e, f} at points {K, H, E, G} correspondingly. The varying point of tangency G

coincides always with the intersection of f with line g = EP, where E is the harmonic conjugate of F with respect to A, B.

There are several other relations hidden in the figure below. Among others the polar of P with respect to the conic passes

through O and J. The figure has a kind of symmetry and could be seen as a construction after the same recipe, but

starting with lines {e,f,g} instead of {a,b,c}. The creation of the conic as envelope of lines (f) is a consequence of the

Chasles-Steiner definition of conics, discussed in Chasles_Steiner_Envelope.html .

their points {K, H} respectively and also tangent to a third line e.To solve it draw line c = KH, find F on line e and its

conjugate E. The conic we are looking for passes through {H,K,E} and two additional points G

intersection points of lines PE and DC, for two auxiliary positions P=P

construct uppon this conic. Thus, view lines {a,b,e} as three arbitrary tangents to the conic. Then for a fourth, variable,

tangent f to the conic, construct point P as intersection of lines AC and BD. Then P moves on a line. This is related to the

fact that the polar of P is line OJ (not drawn). Consequently line (c) coincides, in that case, with the locus of poles of

lines OJ with respect to the conic. But this coincides with the polar line of O, which is (c).

arbitrary point P on (c) we draw PA parallel to b and PB parallel to a and find the intersections of these lines {C, D} with a

and b respectively. Line g is the harmonic conjugate to c with respect to PA and PB and G is the intersection point of g

with line CD. Since PAB is always similar to itself, lines g all parallel for all positions of P. The conic is a parabola with axis

of symmetry parallel to the direction determined by g. The intersection point L of the axis with line (c) is the one for which

line CD becomes orthogonal to the direction of (g). Triangle OCD is circumscribed on the parabola and consequently its

circumcircle passes through the focus V of the parabola. This enables one to find the focus V and gives a way to

construct the parabola tangent to two lines {a,b} at two given points of them {H,K} respectively. An alternative

construction of this parabola is discussed in Artzt.html , see also ParabolaSkew.html .

triangle ABC and line (c) coincides with the polar of A (is parallel to the bisector of angle A). For every point P on that

polar, lines PB, PC intercept on the sides points {I, H} respectively, and line HI is tangent to the excircle, the tangency

point K being at the intersection with line PE.

Harmonic.html

RectHypeRelation.html

Chasles_Steiner_Envelope.html

Artzt.html

ParabolaSkew.html

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