Got stuck at a point -3

While solving the four points on a square Bonus problem I was stuck at a last point (if someone solves this, then I can make a set of steps as an answer to the problem) please try to help.

Consider the following diagram ( $0<\angle PQR < 180\degree$ ) : Let a process $P_1$ on a point $A$ that is on semicircle $QR$ be defined as follows :

Join $\overline{AP}$ and $\overline{AQ}$

Extend $\overline{AQ}$ till it intersects the semicircle $RQ$ at a point B

Join $\overline{RB}$ and extend

Draw the angular bisector of $\angle PAQ$ and let it intersect extended $\overline{RB}$ at point $A'$

Let $A'$ be called as the output of the process

An illustration of process $P_1$ on a point $A$ on the semicircle $QR$ ( $\red{red}$ line is the angular bisector) :

Now imagine that you does this process on every point of the semicircle $QP$ and marks each and every output, what will be the shape made by those outputs? (A circle, A parabola, A square...)

Note:

Any point on the diameter is not counted as a point on the semicircle
If anyone gives the answer with a proof, I will immediately write a solution to the four point in square problem as I am just one solution far to the problem (that is this problem)

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`italics` or `_italics_`	italics
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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Have a look at this, it might help :)

Jeff Giff - 11 months, 2 weeks ago

Wow,how do you make it.. @Jeff Giff

A Former Brilliant Member - 11 months, 2 weeks ago

Sketchbook, iPad. Designed for art, but works well in these situations :)

I had also tried to make this and i also got an approximated circle, but I need a proof for that.

Zakir Husain - 11 months, 2 weeks ago

Yes,math requires proof.I am also trying to prove.

Are QR and QP perpendicular to each other ?? Please reply @Zakir Husain

@A Former Brilliant Member – No they are at an arbitrary angle

@Zakir Husain – Oooooooo ,

Hint: Complete the semicircles. Call the midpoint of arc $PQ$ (the one that the angle bisector will pass through) $M$ , and the two completed circle intersect at another point $Q'$ . You can try to prove that $R,A',Q',M$ are concyclic. (Since $R,Q',M$ are fixed points, this leads to the locus of $A'$ being a circle. A simple angle chasing will prove this. Actually, this is just a special case of Miquel's theorem, or Pivot theorem, so the proof is the same.)

X X - 11 months, 1 week ago

Hope this hint helps, and I'm looking forward to see your solution to the four point in square problem !

Yes,it helps.....

A Former Brilliant Member - 11 months, 1 week ago

I have only 1 doubt, what is the angle between seg QP and seg QR, I know it has to be greater than 0 degrees and lesser than 180 degrees, otherwise getting point B is not possible on the other semicircle, unless the point A coincides with the endpoints of the diameter or is on the diameter, which you have restricted from doing so. Also not greater than 180, because the semicircles would overlap then which I assume is not allowed.

Siddharth Chakravarty - 11 months ago

$0<\angle PQR<\pi$

Zakir Husain - 11 months ago

Yes that is what I meant, thank you! I will work on the proof now and see if I can do it.👍

@Jeff Giff, @Mahdi Raza, @Vinayak Srivastava, @Kumudesh Ghosh, @Aryan Sanghi, @Marvin Kalngan, @Yajat Shamji, @Alak Bhattacharya Try to help

@Kriti Kamal

This one appeared in Terence Tao (yes he!!) 's book in late 90's.

Vishwash Kumar ΓΞΩ - 10 months, 3 weeks ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$