Director circle of ellipse

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There are four completely different definitions of the so-called Apollonius circles:

1. The set of all points whose distances from two fixed points are in a constant ratio

(Durell 1928, Ogilvy 1990).

2. One of the eight circles that is simultaneously tangent to three given circles (i.e., a circle solving Apollonius' problem for three circles).

3. One of the three circles passing through a vertex and both isodynamic points

and

of a triangle (Kimberling 1998, p. 68).

4. The circle that touches all three excircles of a triangle and encompasses them (Kimberling 1998, p. 102).

Given one side of a triangle and the ratio of the lengths of the other two sides, the locus of the third polygon vertex is the Apollonius circle (of the first type) whose center is on the extension of the given side. For a given triangle, there are three circles of Apollonius. Denote the three Apollonius circles (of the first type) of a triangle by

,

, and

, and their centers

,

, and

. The center

is the intersection of the side

with the tangent to the circumcircle at

.

is also the pole of the symmedian point

with respect to circumcircle. The centers

,

, and

are collinear on the polar of

with regard to its circumcircle, called the Lemoine axis. The circle of Apollonius

is also the locus of a point whose pedal triangle is isosceles such that

.

The eight Apollonius circles of the second type are illustrated above.

Let

and

be points on the side line

of a triangle

met by the interior and exterior angle bisectors of angles

. Then the circle with diameter

is called the

-Apollonian circle. Similarly, construct the

- and

-Apollonian circles (Johnson 1929, pp. 294-299). The Apollonian circles pass through the vertices

,

, and

, and through the two isodynamic points

and

(Kimberling 1998, p. 68). The

-Apollonius circle has center with trilinears

(1)

and radius

(2)

where

is the circumradius of the reference triangle.

Because the Apollonius circles intersect pairwise in the isodynamic points, they share a common radical line

(3)

which is the central line

corresponding to Kimberling center

, the isogonal conjugate of the Kiepert parabola focus

.

The vertices of the D-triangle lie on the respective Apollonius circles.

The circle which touches all three excircles of a triangle and encompasses them is often known as "the" Apollonius circle (Kimberling 1998, p. 102). It has circle function

Director circle of ellipse

The locus of points A, from which the tangents to a fixed ellipse (c), make a right angle at A, is a circle with radius

$[0_0]$

Here, a, b, are the major/minor axes of the ellipse. The circle (d) is called [director] circle of the ellipse.

$[0_0]$	$[0_1]$
$[1_0]$	$[1_1]$

The proof follows from the remarks (from the mongraph of Demetrios Bounakis on conic sections, p. 163):
- The foci project on the tangents, to points G, G', H, ... lying on the auxiliary circle c = (K, KI).
- The projections of the foci on the tangent: EG, FH, have |EG||FH| = |FG'||FH| = |FO||FP| = (a+c)(a-c) = b², for c = |EF|/2.
- |AI|² = |AM||AN| = |GE||FH| = b². Hence |AK|² = |AI|²+|IK|² = a²+b² .
For the meaning of the constants, as well as other basic facts on the ellipse, look at Ellipse.html .
Problem: find the locus of points P viewing the ellipse under a fixed angle (phi).

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Director circle of ellipse