Every line through a point C of the directrix of a conic intersects the conic at two points {D,E} such that angle(DAE) is bisected by the polar pC of C. Here A is the focal point corresponding to the directrix i.e. the pole of this directrix.
This is a consequence of the characteristic property of the directrix to have the ratios DXD/DA = e, constant. Here XD is the projection of D on the directrix and e is the eccentricity of the conic. Let F be the intersection of DE with the polar pC of C. Then by the characteristic property of polars
FD/FE = - CD/CE = -DXD/EXE = - DA/EA. This means that F is on the bisector of angle(DAE). Corollary Lines AC and AF are orthogonal. Application To construct a conic (c) knowing two tangents of it, its contact point with one of them
and one of its foci. Look at ConicConstruction.html for the solution.
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A property of the directrix ![[0_0]](DirectrixProperty_files/DirectrixProperty_0_0_0.png)
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