InscribedConicAsEnvelope

Inscribed conic as envelope of tripolars

Let (e) be a conic inscribed in the triangle ABC with perspector at D. Consider line (a') as a variation of side (a), a' being tangent at a point E, different from the traces D₁,D₂,D₃ of D. The following are complementary remarks to the discussion in HyperbolaPropertyParallels2.html , showing that
Every inscribed in a triangle ABC conic (e) is the envelope of the tripolars E₁E₂ of points A'' on its perspectrix.

[1] Apply Pascal's theorem (for quadrangles) to quadrangle ED₁D₂D₃. Points H, E* are intersections of opposite sides and C, E₁ are intersections of opposite tangents. The four points are collinear and make a harmonic division (E₁,C,H,E*)=-1.
[2] Apply Brianchon's theorem (for quadrangles) to quadrangle ABGE₂. Lines AG, BE₂ are diagonals and lines D₂E, D₁D₃ join opposite contact points. The four lines pass through a common point I.
[3] By considering the pole-polar relation for points H and E₂ we deduce that line AG passes through H, line BE₂ passes through E* and triangle HIE* is self-polar.
[4] Let J be the intersection point of D₂H and BE₂. Line D₂H is the polar of B. Line BE₂ is the polar of H. Thus, line BH is the polar of J.
[5] By [1] (E₁,C,H,E*)=-1 and this is equal to (B,E₂,I₂,E*), which from E₁ projects onto (A,G,I,H), which from B projects onto (A,K,E*,A'), A' being the intersection of BH and AE*. These relations imply that A' is the tripole of the tangent at E line E₁E₂.
[6] By the aforementioned discussion A' is on the tripolar of D.

There are several other relations of coincidence and harmonicity of tetrads of points hidden in this figure. For example, points {K,L,E₁} are collinear. See the file InconicsTangents.html for an alternative discussion.

Inscribed conic as envelope of tripolars

See Also