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I am looking for a proof of a discovery as follows:

Let $ABC$ be arbitrary triangle and $(\Omega)$ be an arbitrary circumconic of $ABC$ let $A'B'C'$ is its tangential triangle of $ABC$ respect to $(\Omega)$. Let $BB'$ meet $AC$ at $D$ and $CC'$ meet $AB$ at $E$, let $DE$ meet the circumconic at $F$. A line through $F$ and parallel to $B'C'$ meets $AB$, $AC$ at $H$, $G$ (see Figure) then: $$\frac{HG}{GF}=\frac{\sqrt{5}+1}{2}.$$

Geometric arrangement described in the text

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    $\begingroup$ I think 'amazing' doesn't belong in an MO question, at least not describing your own work. I have edited it out, but of course you may revert if you find it in keeping with MO norms. $\endgroup$
    – LSpice
    May 10, 2021 at 1:45
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    $\begingroup$ @LSpice thank you very much, I open to your suggestions $\endgroup$ May 10, 2021 at 2:03
  • $\begingroup$ An equivalent statement is $HG^2=FG\cdot FH.$ This may be easier to prove. $\endgroup$
    – brainjam
    May 10, 2021 at 3:47
  • $\begingroup$ @brainjam If $\frac{HG}{GF}=\varphi$ $\Rightarrow$ $HG^2=FG.FH$ but $HG^2=FG.FH$ no $\Rightarrow$ $\frac{HG}{GF}=\varphi$ $\endgroup$ May 10, 2021 at 4:41
  • $\begingroup$ This may be a consequence of Poncelet's Porism for $n=3$. $\endgroup$
    – Kapil
    May 10, 2021 at 5:23

1 Answer 1

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Consider generalizing the last line to

Let $I$ be a point on $B'C'$, and let $FI$ meet $AB$, $AC$ at $H$, $G$.

Then the cross ratio of $H,F,G,I$ is the negative golden ratio: $$\frac{HG\cdot FI}{FG\cdot HI}=-\frac{1+\sqrt{5}}{2} = -\phi$$

The original version is the limit of the general version as $I$ goes to infinity.

But the general version is projectively invariant, so we can prove it using coordinates where $A$ is at vertical infinity, the conic is $y=x^2$, and the tangent at $A$ is the line at infinity.

Then the first coordinates are simple: \begin{align} A &= \infty(0,1)\\ B &= (b,\ b^2)\\ C &= (c,\ c^2)\\ A' &= ((b+c)/2,\ bc)\\ B' &= \infty(1,2c)\\ C' &= \infty(1,2b)\\ D &= (c,\ (b-c)^2+c^2)\\ E &= (b,\ (b-c)^2+b^2)\\ F &= ((1-\phi)b+\phi c,\ ((1-\phi)b+\phi c)^2)\\ \end{align} and we can stop computing coordinates there. The graphic below shows the case $b=-1$, $c=2$.

enter image description here

(There is also another solution for $F$, not mentioned in the original question, which leads to switching $\phi$ and $1-\phi$ in the above and the below.)

We want to evaluate the cross ratio $(HG\cdot FI)/(FG \cdot HI)$. Since $I$ is a point at infinity, the $FI$ and $HI$ terms will cancel. Then we can compute the directed quotient $HG/FG$ from the $x-$coordinates, which are $c$ and $b$ for $G$ and $H$. This gives us the desired result: $$\frac{HG\cdot FI}{FG \cdot HI}=\frac{HG}{FG}=\frac{b-c}{(b-c)(1-\phi)}=-\phi$$

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