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beforeAbout an hour ago I was trying to create the harmonic mean using parabola, I haven't managed to get a general answer yet but I got a fairly good answer

enter image description here

Is this theorem known in advance?

I want to link this question of mine with another theorem I mentioned earlier on this site

Harmonic ratio in a parabola and a peaked kissing circle

My question is about a reference that mentions this property if it is already known, and also if someone can create the harmonic mean of two numbers in general using the equivalent of the equivalent then that would be nice, I think I will be able to figure that out during this day, if I succeed in that I will put my method in the answers

Edit: The way I create averages so far in a parabola is to create two lines from focus to parabola and then create an average for those two lengths, look for example at that enter image description here The point $P$ is the middle of the points $M,N$

So I want to generalize my method and make it valid when points $A,B,F$ are not on the same straightness

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  • $\begingroup$ Points $A,F,B$ are on the same line, and $FH$ is parallel to the line $L$ $\endgroup$ Commented May 12 at 6:57
  • $\begingroup$ So are you asking for a proof of this which you found out? Can you please explain what do you mean by "equivalent"? $\endgroup$
    – Gwen
    Commented May 12 at 7:45
  • $\begingroup$ foI don't need to get proof, I can do the proof in more than one way, but proofs are welcome in the answers, as for the wording of the question, it may not be very accurate because it relies on real-time translation from the Microsoft keyboard, but I mean I'm looking for a way to create the harmonic maen of two numbers using the parabola. @Gwen $\endgroup$ Commented May 12 at 8:17

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It seems that this method I'm talking about is just another version of the Cross ladder theorem. enter image description here

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I found another case where the harmonic mean is the horizontal line $FH$, this situation occurs when the points $A,B,M$ are on the same line enter image description here

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