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Now I remembered
my previous question about finding the harmonic mean using a parabola and my answer, which included a second method. That second method inspired me to try more in this configuration to discover more properties, and indeed I arrived at an initial property that I do not know if it was previously known or not, and it can be formulated as follows: If we have a parabola, let $F'$ be the perpendicular projection of the focus point on the directrix of the parabola. Let us consider a chord of the parabola passing through $F'$. The tangents to the parabola at both ends of this chord will intersect at a point which belongs to the straight line parallel to the directrix and passing through the focus. enter image description here Is this property already known ? Also, are there more known properties of this configuration ?

Also, please mention the references on this subject if it is known in advance.

Jean Marie
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    Interesting. I don't know if it's novel, but the role of the focus and directrix (or "guide") isn't unique: If $P$ is any point on the axis of symmetry, and $AB$ is a chord whose extension passes through $P$, then the tangents at $A$ and $B$ meet at a point whose projection onto the axis is the reflection $P'$ of $P$ in the parabola's vertex. (Easy case: When $P$ is the vertex.) ... Also, a variant of this property holds for non-parabola conics, although $P$ and $P'$ are more naturally related via the conic's center than its vertex. (This makes me think that the property is known.) – Blue Jul 07 '24 at 06:13
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    Thank you @blue, you are right, this property is always true with respect to the electrode and the electrode, I will try to use the same idea for the rest of the properties – زكريا حسناوي Jul 07 '24 at 07:15
  • Pay attention to your translator : the word "electrode" you have used in the previous comments has no mathematical meaning. It is an electrical device... – Jean Marie Mar 21 '25 at 17:26
  • Proof : In fact, this particular property is a direct consequence of the so-called "La Hire theorem" : if the polar line of a point $P$ with respect to a conic curve passes through $Q$, then the polar line of $Q$ passes through P. See p. 14 of this document where a circle replaces the parabola because in projective geometry, a parabola can be replaced by a circle for any proof. Here $P=F'$ and $Q=M$. – Jean Marie Mar 21 '25 at 18:08

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Now I discovered another property related to this configuration The center of the circle passing through the points A, B, M belongs to the axis of symmetry of the parabola

enter image description here

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And this is another feature that I have come up with now related to this configuration

The line passing through M and perpendicular to AB must pass from a fixed point which is the image of point F' relative to point F enter image description here

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And this is another feature I've just come up with about this configuration.

The two straight lines AF, BF are made with parabola guide isosceles triangle enter image description here

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