## 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.

### See Also

Brianchon.html
HyperbolaPropertyParallels2.html
InconicsTangents.html
PascalOnQuadrangles.html
TrilinearPolar.html
Trilinears.html

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