[alogo] A property of the directrix

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.

[0_0] [0_1]
[1_0] [1_1]
[2_0] [2_1]
[3_0] [3_1]

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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