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About ten days ago I discovered three beautiful properties of a parabola with three tangents drawn using the GeoGebra program. I would like to get proof of these properties and also know if any of them have been discovered before or not.

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We have a parabola, and we chose three points from it, $A$, $B$, and $C$. We drew the tangents to the parabola in them, which enclose the triangle $∆MNP$, so the area of triangle ABC will be equal to twice the area of triangle $∆MNP$. In other words $T=2S$

Another beautiful property is achieved in the form: $\frac{FA×FB×FC}{FM×FN×FP}=1$

Also, the lines $AN$, $BM$, and $CP$ converge at one point

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  • $\begingroup$ The first property is known : see here. I didn't know the second property nor the third. $\endgroup$
    – Jean Marie
    Commented Nov 22, 2023 at 20:09
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    $\begingroup$ A little remark : you don't need figure 1. Figure 2 is enough. $\endgroup$
    – Jean Marie
    Commented Nov 22, 2023 at 22:12
  • $\begingroup$ A result that could be helpful : Lambert's theorem asserting that the circumcircle of triangle MNP passes through focus $F$. $\endgroup$
    – Jean Marie
    Commented Nov 22, 2023 at 22:43
  • $\begingroup$ Another useful result : the fact that the orthocenter of triangle $MNP$ belongs to the directrix of the parabola can be found in this answer of mine here. $\endgroup$
    – Jean Marie
    Commented Nov 23, 2023 at 1:55
  • $\begingroup$ Thank you very much, yes I know Lambert's theorem, but I did not know that the first property was previously discovered, but it is expected because it is an elementary property of the parabola $\endgroup$ Commented Nov 23, 2023 at 2:03

1 Answer 1

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Let's prove the third property in the special case of a circle: as projective transformations preserve collinearity, the result holds for any conic section.

The proof follows from the converse of Ceva's theorem applied to triangle $PMN$: lines $PC$, $AN$, $BM$ concur if:

$$ {BN\over BP}\cdot{AP\over AM}\cdot{CM\over CN}=1. $$ But that is trivially true, because $$ BN=CN,\quad BP=AP,\quad AM=CM. $$

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