## Remarks on Thales theorem

[ 1 ] Consider three lines {a,b,c} passing through point O. Intersect these lines with a transversal line (e) at points
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 harmonic conjugate of F with respect to {A,B}. The two
cases are discussed respectively in Thales.html and Harmonic.html .

[ 2 ] Points E are located on the harmonic conjugate line of c=OP with respect to the pair of lines (a,b). More general, if a
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).
[ 3 ] Working on the same figure with the same two cases. One could define a map as follows: for each line (f), parallel to
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.
[ 4 ] Next we modify slightly the previous picture by allowing line (c) not to pass through the intersection point O of
(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 .

[ 5 ] The above figure can be seen also as a solution to the problem: To construct a conic tangent to two lines {a, b} at
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 G1, G2, constructed as
intersection points of lines PE and DC, for two auxiliary positions P=P1, P=P2 of P on line (c).

[ 6 ] Yet another viewpoint for the same figure is to consider the conic as a primary element and the rest of the figure as a
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).

[ 7 ] An interesting specialization of the previous case occurs when e is identified with the line at infinity. Then for an
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

[ 8 ] Another interesting specialization occurs when the conic is a circle. This circle then is an escribed circle of the
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.