ParabolaPropertyII

Parabola properties II

On basis BC of triangle ABC select a moving point D and draw the two circles {ABD} and {ADC}.
1] The line of centers of these circles envelopes a parabola with focus at A and directrix BC.
2] The common tangents of the two circles intersect at a point K moving on a line L.
3] The angle of the two tangents at K is constant and independent of the position of D.
4] Line L and the medial lines of sides {AB, AC} are tangents of the parabola.
5] The whole moving figure remains similar to itself. In particular triangle ADK remains similar to itself.
6] The angle formed at K (moving on tangent L) from KA and the other tangent KB' is constant.
Thus, from every point K moving on a tangent L the angle between the other tangent from K and the focal radius KA is constant.
7] The same is valid for the angle formed by line KA and each of the common tangents to the two circles.
8] Line L intersects BC at a point M on the medial line of the bisector AI at A and is parallel to AI.
This implies that the two tangents from points M on the directrix of the parabola are orthogonal.
9] Triangle C'B'P whose vertices are the centers of the circles and the center of the circumcircle of ABC has a circumcircle which passes through point A which is the focus of the parabola.
10] Each of the common tangents of the two circles envelopes a parabola with the same focus A.
11] Point M has the following remarkable properties.
  a) MI is the bisector of angle UMP.
  b) The triangle formed by the tangents {PQ,MI,PR} is isosceles with angle at P equal to (A).
  c) Triangles MQP and MPR are similar.
  d) Point M is aligned with the contact points Q,R which define the polar QR of P.
  e) The polar of M is line OP which passes from the focus A and the contact point T of the medial line of AI.

[1] follows from the fact that the medial line KG, with GD orthogonal to BC, has GA=GD, thus G is on the parabola c with the stated properties. It is easily seen that the other than G points of KG are not on this parabola, hence this line is tangent to c. Particular positions of line KG are the medial lines of the sides AB, AC.
[2-7] If A' and B' are the centers of the two circles then ABA' and AB'C are similar triangles and AA'B' is similar to the triangle of reference ABC. Thus the whole figure varying with D remains similar to itself all the time.
This implies that the ratio KA'/KB' is constant hence point K is moving also on a tangent L to parabola c.
This results from the fundamental property of parabola tangents (its inverse actually) discussed in ParabolaProperty.html .
The ratio intercepted on a variable tangent (KG) by three others {PQ, PR, L} is constant.
[8] This follows from the similarity-invariance of the moving figure, by which angle AKG is constant and equal to (B-C)/2.
The second statement follows from the equality OS=OA and SM=MA, where O is the contact point of L and S its projection ont the didrectrix. By the isosceles AMI, AM=MI, which implies that AS is parallel to the medial line of AI.
The claim follows from this. Since every point M on the directrix can be considered as a point constructed from such a configuration the orthogonality of tangents from points of the directrix follows also immediately.
[9] follows from the angles at A and P of the quadrilateral PB'AC'. At A the angle is constantly equal to (A) and at P is either (pi-A) or also (A).
[10] follows from [2-7] and the Newton generation of a parabola by a varying angle (see ParabolaNewton.html ).
[11] (a) follows from the orthogonality of MI to MO (see Harmonic_Bundle.html (3)).
It follows that the polar QR of P is the reflected to MI of line MP. The isosceles property follows from the orthogonality of sides of the triangle to the sides and the bisector of ABC at A. This proves (b) and from this follows immediately (c).
The proofs of (d,e) follow by observing that since M is on the polar of P, the polar of M which is OI will pass also from point P.
Besides the polar of M passes also through A, since M is contained in the polar of A which is the directrix BC.
Corollary For every isosceles circumscribing a parabola the basis and the polar of the apex meet on the directrix.

Parabola properties II

See Also