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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, and we set the perpendicular projection of the focus point on the guide of the parabola. Then we draw a chord of the parabola that passes through that point. The tangents to the parabola at both ends of this chord will intersect at a point in the straight line parallel to the guide of the parabola and passing through the focus. enter image description here Is this property known in advance? Also, are there more known properties of this composition?

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

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    $\begingroup$ 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.) $\endgroup$
    – Blue
    Commented Jul 7 at 6:13
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    $\begingroup$ 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 $\endgroup$
    – زكريا حسناوي
    Commented Jul 7 at 7:15

3 Answers 3

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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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  • $\begingroup$ This property is not true for the pole and the pole in general $\endgroup$
    – زكريا حسناوي
    Commented Jul 7 at 7:51
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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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  • $\begingroup$ This property is also generalized for the pole and the pole always $\endgroup$
    – زكريا حسناوي
    Commented Jul 7 at 7:52
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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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  • $\begingroup$ This property is always true for the pole and the pole. $\endgroup$
    – زكريا حسناوي
    Commented Jul 7 at 7:52

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