[alogo] Parabola Chords

Consider the parabola with directrix line (a) and focus F. The following properties are valid.
[1] The line DG joining D to the middle G of PP' is parallel to the axis of the parabola.
[2] The intersection point E of the parabola with DG is the middle of DG.
[3] The tangent at E is parallel to the chord PP' and QQ' is the half of PP'.
[4] The middle points G of all chords PP', parallel to a given direction e are on a line DG parallel to the axis of the parabola.
[5] The previous line DG is the parallel to the axis going through the tangent point E with a line parallel to the direction of PP'.
[6] The line HF is orthogonal to the chord direction.
[7] The segments on the chord UV, U'V' between the parabola and the tangents are equal.
[8] The segments UW, U'W' are also equal.
[9] Triangles DFP and DFP' are similar and DF2 = FP*FP'.
[10] For triangles like DQQ', formed by tangents, their circumcircle passes through the focus F.
[11] The Simson-Wallace line of F with respect to triangle DQQ' is the tangent at the vertex of the parabola. The orthocenter of DQQ' is on the directrix of the parabola.
[12] DF is the bisector of angle(PFP'), which is twice the angle(PDP').
[13] Tangents DP, DP' are orthogonal exactly when D is on the directrix (a) of the parabola. Then PP' passes through F. Inversely, if PP' passes through F then the tangents at P, P' are orthogonal and meet on (a).
[14] DF is the symmedian of triangle DPP' from D.

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

[1] Because PD, PD' are medial lines of AF, A'F, D is the circumcenter of the triangle AFA'. D projects on the middle H of AA'.
[2] Apply [1] to the tangent pairs QE, QP and Q'E, Q'P'. R, R' are the middles of GP, GP' hence QQ' joins the middles of PD, DP'.
[3] Follows from [2].
[4] DG passes through the fixed point E, where the tangent is parallel to the direction e. Its direction is orthogonal to (a).
[5] See above.
[6] The tangent at E is medial line of HF.
[7] UU' and VV' have a common point as their middle.
[8] Follows from the previous.
[9] Follows from the facts: angle(DFP') = angle(DFP) = angle(DAP). angle(S'DF) = angle(S'SF) = angle(HAF) = angle(APS) = angle(SPF). The second equality because of the cyclicity of DSFS' and the parallelity of SS' to AA'.
[10] Follows from the previous, since angle(Q'FS') = angle(QFS) => angle(QFQ')+angle(QDQ') = pi. Notice also that triangle QFQ' is similar to PFD, DFP', since FQ, FQ' are the medians of the two similar triangles.
[11] Follows from the fact that the Simson-Wallace line of DQQ' with respect to F passes through points {S,S'}. The locus of these points is the tangent at the vertex. The statement on the orthocenter follows from the characteristic property of the Steiner line (see SteinerLine.html ).
[12] Follows easily from [9]. [13] is a consequence of [12] since angle(ADA') = angle(PFP').
[14] Follows also from [9], since the ratio FS/FS = DP'/DP, which is characteristic for the symmedian from D. Notice that [12] and [14] identify F with a vertex of the second Brocard triangle and the parabola with one of the Artzt parabolas (first kind) of triangle DPP'
These properties are used in the discussion about the determination of the parabola in oblique axes (see ParabolaSkew.html ).
Look at the file ParabolaArea.html for the Archimedean method to calculate the area of a parabola sector. The file Miquel_Point.html contains a discussion on the parabola tangent to four given lines.

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