Jtdylktuy

I was messing around on GeoGebra when I discovered a mysterious hyperbola in a quadrilateral, among other really fascinating theorems. I'm not $100%$ certain what I have found is true, as I have no mathematical proof (that's what I'm here for). However, I have tried many, many different configurations and these theorems always seem to hold.

Let $ABCD$ be a quadrilateral (not necessarily convex) with points $P=ADcap BC$, $Q=ABcap CD$, and $R=ACcap BD$. Let $E$, $F$, $G$, $H$, $I$, and $J$ be the midpoints of $AC$, $BD$, $AB$, $CD$, $AD$, and $BC$, respectively.

Here's what I need help proving:

1. 
   $E$, $F$, $G$, $H$, $I$, $J$, $P$, $Q$, and $R$ are conconical on a conic $K$.

2. 
   $(PIJ)$, $(QGH)$, and $(REF)$ concur at a point $O$, and $O$ lies on $K$.

There are also some theorems with Miquel Points.

Define the first Miquel point $M_{1}$ as the point of concurrency between circles $(PAB)$, $(PCD)$, $(QAD)$, and $(QBC)$.

Similarly, define the second Miquel point $M_{2}$ as the point of concurrency between circles $(RAD)$, $(RBC)$, $(PAC)$, and $(PBD)$.

Lastly, define the third Miquel point $M_{3}$ as the point of concurrency between circles $(RAB)$, $(RCD)$, $(QAC)$, and $(QBD)$.

Here's what I need help proving:

3. 
   $triangle M_{1}ABsimtriangle M_{1}DCsimtriangle M_{1}IJ$.

4. 
   $M_{2}$, $M_{3}$, $E$, $F$, and $R$ are concyclic, and $M_{2}EM_{3}F$ is harmonic.

5. 
   $M_{1}$, $R$, and $O$ lie on a line.

Please feel free to help prove even just one of the above theorems. It'll help a lot.

I have relatively little background in any area of geometry that deals with conics, so if you give a hint/proof to the existence of the hyperbola, please explain with a fair amount of detail. I really appreciate it!

Thanks for all the help.

Edit $1$: I think I've managed to prove all of the above theorems except for the existence of $O$, the collinearity of $M_{1}$, $R$, and $O$, and the existence of the 10-Point conic through $O$. I'm still open to alternative proofs, however, so feel free to give a solution to any of the above theorems.

Too long for a comment ...

I've used Mathematica to verify the result (barring pathologies). So that's a good thing. :)

I'm seeking a clean solution to the problem. In the meantime, I can't help but comment on the presentation of the problem.

There's a lot of stuff for a reader to sift-through here. Your presentation would benefit greatly from some re-organization. For instance ...

1. Let $square ABCD$ be a (convex) quadrilateral. (I don't believe convexity is actually required.)


2. Define $E$, $F$, $G$, $H$, $I$, $J$ as the midpoints you describe.


3. Note that the six midpoints lie on a common conic, $K$. (Proof required.)


4. Define $P$, $Q$, $R$ as you describe. (Care should be taken with points at infinity.)


5. Note that $P$, $Q$, $R$ lie on $K$. (Proof required.)

Actually, (3) and (5) are "known", since $K$ is Bôcher's nine-point conic (defined by exactly those nine points). You may wish to see a proof, which is fine. In any case, all but one of your ten points are quite-uncomplicated beasts, and their conconicality should be at least plausible to the reader. Discussing them first gets the easy stuff out of the way.

Then, to embark on constructing the curious point $O$ and proving that it, too, lies on $K$.

This is where I lose the thread a bit.

You begin by introducing Miquel points, use those to determine cyclic quadrilaterals, and then identify $O$ as the point of concurrence of the quadrilateral circumcircles. Fine. Yet ... Theorems 2.x effectively say that the quadrilateral circumcircles are simply the triangle circumcircles $bigcirc PIJ$, $bigcirc QGH$, $bigcirc REF$.

So, I would proceed thusly.

6. Note that $bigcirc PIJ$, $bigcirc QGH$, $bigcirc REF$ concur at a point ($0$). (Proof required.)


7. Note that $O$ lies on $K$. (Proof required.)

Done!

As before, since all the defining elements are uncomplicated, the conconicality property would seem to be plausible, and proof wouldn't seem particularly daunting. (Mathematica made light work of this.)

So, if the goal is simply to get to $O$ and show that it's on $K$, the Miquel point stuff seems like a distraction ... and a great deal of extra effort.

That said, if you're actually interested in the Miquel points and their concyclic/harmonic/collinear properties, or think that those properties might be useful in establishing (6) and (7), it's worth mentioning them, but separately from the above and as optional considerations. So, we have

8. Define the $M_i$ as you describe. (Proof of existence required.)


9. Note various similarities (Theorems 1.x). (Proof required.)


10. Note the concyclic properties (Theorems 2.x). (Proof required.)


11. Note the harmonic properties (Theorems 2.x). (Proof required.)


12. Note the collinear properties (Theorems 4.x). (Proof required.)

Restructuring your question this way makes it clearer to the reader what is and isn't required to get to the result you seek.

By the way ... As a matter of style, it isn't really necessary to list every similarity, concylicality, and collinearity as a separate "Theorem". ("Lemma" is a better word, anyway, but be that as it may ...) Simply state one such result, since the others will follow from similar arguments. For example: "$M_1$, $R$, $O$ lie on a line. (Likewise, $M_2$, $Q$, $O$, and $M_3$, $P$, $O$.)"