Poncelet's angle relations concern the angles between the tangents {QM,QN} and the focal radii {QF,QF'} of an ellipse/hyperbola. Here I discuss the case of the ellipse. The case of hyperbola can be handled analogously.
Let line L be the directrix of the ellipse c, corresponding to its focus F (i.e. L is the polar of F).
[1] For any point P on L and line PMN intersecting (c) at {M,N} angle(MFN) is bisected by the polar LP of P.
[2] The polar LP of P is orthogonal to line PF.
[3] For any point Q outside the ellipse line QF bisects angle(MFN), {M,N} being the contact points of the tangents from Q.
[4] angle(NQF') = angle(MQF), F' being the second focus of the ellipse.
Since P is on the polar of F, LP passes through F. By the polar property P'M/P'N=PM/PN=MM'/NN'. Besides by the fundamental property of the directrix MM'/NN'=FM/FN. Thus {FP,FP'} are the bisectors of the angle(MFN).
This proves [1,2]. Property [3] is a consequence of the other two, since for any point Q we can define the corresponding P on the directrix through its inetersection with the polar MN of Q.
To see [4] take the symmetrics {N'',M''} of the second focus F' with respect to the tangents {QN,QM} and consider the angles at Q: angle(N''QF)=2u+w = angle(FQM'') = v+angle(MQM'')=v+(v+w). It follows 2u=2v proving the last claim. Use was made of the fact that the symmetrics of one focus with respect to tangents together with the other focus and the contact point are aligned (see Ellipse.html ).
An immediate consequence of the previous properties is the following one concerning the portion of a tangent between two other tangents of a conic.
The segment BC intercepted by two tangents {DE,DG} of a conic on a third tangent is viewed from a focus F under a constant angle equal angle(EFG)/2, where {E,G} are the contact points of the tangents from D.
This can be seen also as a means to produce a conic by rotating a fixed in measure angle (BFC) about a fixed point (F):
Given an angle XDY and a fixed point F we intersect sides DX, DY of the angle with the sides of another angle BFC of fixed measure but turning around F. Then lines BC resulting by the various positions of angle(BFC) envelope a conic with one focus at F.
The conic can be easily constructed from the given data. In fact, for C coinciding with D we get B coinciding with G. Analogously for B coinciding with D we get C coinciding with E. We can easily construct the conic with focus at F and given tangents at G and E and having lines BC as tangents.
Notice that the kind of the conic produced by this procedure depends on the sum of the angles (fixed + turning)
S = angle(BDC) + angle(CFB).
When S is less than two right angles the resulting conic is an ellipse (the case shown above). When S is equal to two right angles then the resulting conic is a parabola and when S is greater than two right angles the conic is a hyperbola.
The figure below illustrates the results of the first section for the case of parabolas. The figure speaks from itself. The same arguments apply to prove the corresponding results. Note in particular the similarity of triangles {FNQ, FQM, FRS}.
In the case of parabola, fixing the two tangents {QN,QM} and varying the third RS produces triangles FRS which remain similar to each other all the time. This does not happen for the other cases of conics (ellipses and hyperbolas).
The figure is worth watching and discovering additional relations. For example that the circumcircle of triangle QRS passes through the focus F of the parabola. The equality of angles angles at Q suggested by the yelow marks is the translation of [4] for this case.
Considering the envelope aspect of section 2, the parabola results when the turning angle (angle(RFS)) is complementary to the fixed one (angle(RQS)). In other words, when the quadrilateral FRQS is cyclic.
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