Thanks for yr replies.

{I realised the foll only AFTER I had drafted the body of the mail, and when I proceeded to attach ! :
Could not attach since the site limits to max of 20 KB only !
The 2 (sacnned) pdf files are each about 1.1 MB -> about 310 KB in jpg format.
What do I do ?
Is there an alternate way to send you the 2 Construction figures ?

Anyway, pl read on the foll, and let me know how to send the 2 figs.}

***********

The main mail :

1) Before I went to check the CTK site again for any replies, I had myself figured out the way – mainly with the help of the 2 diagrams in : https://mathworld.wolfram.com/Arbelos.html

Pl see attached construction : “Arbelos - My Constrcution Case 1A.pdf ”
{The labels may be different wrt BOL5. For eg, “X” that we coined earlier is “D” in my current attached figure.}

***********

2) Yes, AD (earlier called AX) is the “Power Radius” of the “Ref Circle of Inversion” – wrt points B and C.
In the case of the attached fig, this radius AD = sqrt (12 * 18) = 14.69693846 units.

But my feeling is that this circle with radius AD (or the earlier AX), may NOT be the Reference Circle of Inversion when it comes to circle pairs O3’ <-> O3 and W1 <-> W1’.

Pl refer the construction details with all calculations in :
“Arbelos - My Constrcution Case 1B.pdf”
(this is more detailed – see this “1B version” only AFTER you have seen the “1A” version.)

Consider for eg, points P and R in O3’ ; the inverses of these 2 points – wrt the above Ref Circle with radius AD = 14.697 units – do not seem to map to points in the current O3 circle (O3 center is XB) ; O3 is tangential at 3 points YB1, YB2 and YB3 to the 3 circles.
That is the real dispute here.

***********

3) Yes, in my specific example case, W3B (W3B – B is for Bottom) and O3’ are indeed tangential ! I have checked with calculations for the coordinates of “S”.
In other cases, they will intersect.

***********

PS : In case, you are wondering what YB1, YB2, YT3, YT1 etc are, the logic for labeling is quite simple :
T is for the Top portion, and B is for Bottom ;
W3T is a Top mirror image of W3B – it (ie, W3T) is tangential to the 3 circles at YT1, YT2 and YT3. …

The form that comes up after clicking on the icon : "Click to email alexb" does not have a facility to "attach" ? Atleast, I could not see any "Attach" icon.

So, how do I send you the 2 pdf files ?

Ans : I went to the index page of the website where I found alexb@... (purposely not showing an email id in this public forum).
I then mailed to this email id the 2 pdf files using my Yahoo email id.

{To get yr email id, I proceeded thus :
https://www.cut-the-knot.org/index.shtml -> Clicked on "Alexander Bogomolny" : https://www.cut-the-knot.org/MailNotificationPage.shtml which states :
("Use my email address only if there is indeed something very private you want to communicate.")}

*****************

Pl acknowledge proper receipt of this email. Thanks.

And while acknowledging, pl do provide your exact email id explicitly - so that I can send you the 2 pdf files again.

*****************

But, in the meantime, I have already posted another message today (11th Aug) at 1 PM - have you not yet seen this - for "moderation" ?

I have once again copied the whole message of 1 PM below.

{While "moderating", if you find my 1 PM message, pl display the 1 PM posting, and you may remove this 6 PM posting since this 6 PM msg will then be redundant.}

*****************
*****************

Sub : RE: Arbelos : Proof for R = r1 * r2 / (r1 + r2)

Alexb, 11th Aug, 2008, 1 PM

I had a relook at the same construction, and I now feel that the small discrepancies between the measured and the calculated values for Inversion checks wrt O3’ <-> O3 and W1 <-> W1’ may perhaps be due to manual construction errors (the pencil and pen lines unfortunately have some thickness !), scale errors and graph errors.
Very initially, I also made the mistake of considering the center points of the circles for Inversion checks.
So, for now, let us take the “Ref Circle of Inv” point as resolved ie, the circle with radius = AT = AD is the Ref Inv circle for the “all” the circles in the “whole” diagram.

***************

Even though I could construct and proceed, the question of “proving” that the “value” of the radius “R” of W1 and W2 is = r1 * r2 / (r1 + r2) still remains.

To start with, assume that we do not know the value of R magnitude ie, we have not yet proved that R = r1 * r2 / (r1 + r2).
Pl note that if we do not know this “value of R”, we do not the x coordinates yet, of the center of W1. For eg, in my fig, we do not yet know the distance B–W1P (2 units in my sketch case).
Ironically, I had started the whole construction with this first assumption that B–W1P is R = 2 units. But, what if we do not yet know this offset value ?

Yes, it'seems quite intuitive that there may be a reciprocal relationship between R and r1 & r2 ie, 1/R = 1/r1 + 1/r2. But how do we “prove” this ?

***************

I tried to gather what we know of W1 :

Let (a, b) be the center of W1, and let the radius be R. At this point, we only know r1 = say, 6 units, and r2 = say, 3 units.
Let W1 be defined by : (x – a) ^2 + (y – b)^2 = R^2

1) W1 is tangential to the line BED at some point E (x1, y1) – where x1 = 12 (we do not yet now the y1 coord.)
So, (12 – a) ^2 + (y1 – b) ^2 = R^2
ie, y1^2 – 24a – 2*b*y1 = R^2 – a^2 – b^2 – 144

2) At G (x2, y2), W1 is tangential to the circle with dia = AB ie, : (x – 6) ^2 + y^2 = 6^2 = 36 <= r1^2>
From Circle with dia = AB : x2^2 + y2^2 – 12*x2 = 0
And, from W1 : x2^2 + y2^2 – 2*a*x2 – 2*b*y2 = R^2 – a^2 – b^2

3) At F (x3, y3), W1 is tangential to the circle with dia = AC ie, : (x – 9) ^2 + y^2 = 9^2 = 81 <= (r1 + r2) ^2>
From Circle with dia = AC : x3^2 + y3^2 – 18*x3 = 0
And, from W1 : x3^2 + y3^2 – 2*a*x3 – 2*b*y3 = R^2 – a^2 – b^2

But, how do we proceed next ?

I still feel that there must be a far more easier method to prove the value of R in relation to r1 and r2, mainly because of the common perpendicular line BEDJ – common to the 4 circles : dia AB, O2, W1 and W2 ; it is perhaps too simple ! – so simple that I am missing it. What is the missing link ?

Rgds

Sundar Krishnan

*****************

PS : I have observed this again :
Any comment placed within <...> does not get displayed ??
So, I had to put such comments under {...}.
Pl check.

*****************

I received two messages of yours at my private email. This mean you found a way to send me the 2 pdf files.

Please do not abuse this way of communication.

>PS : I have observed this again :
>Any comment placed within <...> does not get displayed ??
>So, I had to put such comments under {...}.

Please read

https://www.cut-the-knot.org/cgi-bin/dcforum/forumctk.cgi?az=help&cat=d&question=14

https://www.cut-the-knot.org/Curriculum/Geometry/BookOfLemmas/BOL5.shtml#explanation

you can observe a derived identity

AC·CB = AB·HE

Which means

2r₁·2r₂ = (2r₁ + 2r₂)·2R

In other words,

1/R = 1/r₁ + 1/r₂.

Thanks so much for the proof. Something told me it was simple, and I was missing it.
Thanks again.

***************

I sent those 2 pdf files only on yr request - yr msg dt Aug-10-08, 01:16 PM (EST).

***************

Box brackets' pair (different from plain brackets : (...), and flower brackets : {}) - either does not get displayed with all the contents in between, or gets displayed as <...> !

For eg, in my msg of 1 PM / 6 PM of 11th Aug, the Box Brkts get displayed as <...> :
y^2 = 6^2 = 36 <= r1^2>

I did not mean <= less than !

Using Flower Brackets instead of Box Brackets, this is what I wanted :
y^2 = 6^2 = 36 {= r1^2}
OR
y^2 = 6^2 = 36 {ie, = r1^2}

You may want to check on the character syntax in the S/w.

***************

Is there a better way to show text features such as highlights, different colors, superscript, underscores etc in this site ? Also, pl let me know if there are any on-board features in this site to write out mathematical expressions like Integrals, square-roots etc - something similar to, or better than the nrich site ?

***************

Links to some "Farlex Dictionary ?" get automatically created in the Preview of messages - even though I have never intentionally created any ref links. For eg, blue, underscored links show up even for terms like :
"was !", "pdf files", "flower" etc.

Is there a way to avoid unintentional links ?

***************

As I requested earlier, you may now delete the redundant message dt
Alexb, 11th Aug, 2008, 1 PM {Aug-11-08, 10:04 AM (EST)}

Please read the documentation I pointed out previously. Square brackets are used for HTML tags instead of angular brackets.
>
>Is there a better way to show text features such as

Just use square brackets.

>Links to some "Farlex Dictionary ?" get automatically
>created in the Preview of messages - even though I have
>never intentionally created any ref links. For eg, blue,
>underscored links show up even for terms like :
>"was !", "pdf files", "flower" etc.

This is just a way to make a little money. This is advertisement of sorts.

The HTML tags can be classified as "Angular Brackets" atleast for our disc ; let us keep these angular brackets out of consideration for now.

Any text that appears within Square (or Box) Bkts does NOT show up in the preview - I have seen this happen atleast 3 times.

Also, as I have already indiacted, if in the foll expression, you replace the Left and Rt Flower Bkts { & } by Left and Rt Square bkts, it gets dispayed instead with Angular Bkts ! :

y^2 = 6^2 = 36 {= r1^2}
// replace { & } in the above expr by Left and Rt square bkts and try

I had a relook at the same construction, and I now feel that the small discrepancies between the measured and the calculated values for Inversion checks wrt O3’ <-> O3 and W1 <-> W1’ may perhaps be due to manual construction errors (the pencil and pen lines unfortunately have some thickness !), scale errors and graph errors.
Very initially, I also made the mistake of considering the center points of the circles for Inversion checks.
So, for now, let us take the “Ref Circle of Inv” point as resolved ie, the circle with radius = AT = AD is the Ref Inv circle for the “all” the circles in the “whole” diagram.

***************

Even though I could construct and proceed, the question of “proving” that the “value” of the radius “R” of W1 and W2 is = r1 * r2 / (r1 + r2) still remains.

To start with, assume that we do not know the value of R magnitude ie, we have not yet proved that R = r1 * r2 / (r1 + r2).
Pl note that if we do not know this “value of R”, we do not the x coordinates yet, of the center of W1. For eg, in my fig, we do not yet know the distance B–W1P (2 units in my sketch case).
Ironically, I had started the whole construction with this first assumption that B–W1P is R = 2 units. But, what if we do not yet know this offset value ?

Yes, it'seems quite intuitive that there may be a reciprocal relationship between R and r1 & r2 ie, 1/R = 1/r1 + 1/r2. But how do we “prove” this ?

***************

I tried to gather what we know of W1 :

Let (a, b) be the center of W1, and let the radius be R. At this point, we only know r1 = say, 6 units, and r2 = say, 3 units.
Let W1 be defined by : (x – a) ^2 + (y – b)^2 = R^2

1) W1 is tangential to the line BED at some point E (x1, y1) – where x1 = 12 (we do not yet now the y1 coord.)
So, (12 – a) ^2 + (y1 – b) ^2 = R^2
ie, y1^2 – 24a – 2*b*y1 = R^2 – a^2 – b^2 – 144

2) At G (x2, y2), W1 is tangential to the circle with dia = AB ie, : (x – 6) ^2 + y^2 = 6^2 = 36 <= r1^2>
From Circle with dia = AB : x2^2 + y2^2 – 12*x2 = 0
And, from W1 : x2^2 + y2^2 – 2*a*x2 – 2*b*y2 = R^2 – a^2 – b^2

3) At F (x3, y3), W1 is tangential to the circle with dia = AC ie, : (x – 9) ^2 + y^2 = 9^2 = 81 <= (r1 + r2) ^2>
From Circle with dia = AC : x3^2 + y3^2 – 18*x3 = 0
And, from W1 : x3^2 + y3^2 – 2*a*x3 – 2*b*y3 = R^2 – a^2 – b^2

But, how do we proceed next ?

I still feel that there must be a far more easier method to prove the value of R in relation to r1 and r2, mainly because of the common perpendicular line BEDJ – common to the 4 circles : dia AB, O2, W1 and W2 ; it is perhaps too simple ! – so simple that I am missing it. What is the missing link ?

Rgds

Sundar Krishnan