= 0,3333333333π rad

= 1,0471975512 rad

Dear @Liudmyla Hetmanenko

That's a very hard question.

Indeed, it is about the future of human kind :-) Literally treated this question cannot be answered by a human being.

Fortunately we do not have to be rigorous if it is impossible and we can focus our efforts on the geometrical problem presented within the question.

An attractive conjecture appears that without additional conditions, at any positions of the hords with common point inside the big circle there are uniquely determined 4 circles inscribed as it is drawn on the picture. I think that this guess should explained in the first stage. If this can be done then this will be full answer. But even without this full analysis we can say that any number from the open interval (0,π) may be the measure of the angle APD (see PS. below)+

In such circumstances a didactic question may be posed: How can we modify the problem (say, by adding additional requirements upon the relative positions of the centers) to obtain algebraic equations for other parameters (say radia of the circles).

There are many possibilities.

PS. The answer by @Ruslan Pozynkevich is only one of possibilities. I think it is easy to see that with position of P in the center O with any angle between the hords we can inscribe the four circles.

Dear @Liudmyla Hetmanenko

Studying the picture attached to your question I become suspecting that your problem has an unexpressed condition that the pairs of circles 1&4 and 3&4 possess common points where they are tangent to the lines CD and AB, respectively. This would specify the problem to really interesting case. Can you, please confirm if this was your wish.

Best regards, Joachim Domsta

Dear Liudmyla Hetmanenko ,

I appreciate and highly esteem your efforts in continuing so much valuable activity in education in math, despite your everyday life is dramatically and tragedly continuously perturbed by the unjustified war agression of Russia. This motivates me in contributing to this exercises by taking part in discussing your creative and very interesting problems of geometry. Wishing you and your friends (personal as well as all other colleagues and students) that despite such severe troubles the teaching and learning will continue as desired, I dare to present my proposal of an answer to the geometric part of this thread.

In order to have a well defined issue Ie have taken under considerations only the scenario confined by the condition: every pair of circles, 1&4 and 3&4 possesses common point where they are tangent to the lines CD and AB, respectively. Here is my answer:

For every pair of positive numbers R and r, r

Joachim Domsta look the picture, pls

You are right, I have corrected the question. My mistake. Thank you for your mathematical vision of the problem

Joachim Domsta Hello Joachim! I don't have a valid approach for that, but I try in the following to help with some remarks:

1. points O1, P, O3 are on a same line( are colinear), line which bisects the angle APD

points O2, P, O4 are on a same line( are colinear), line which bisects the angle APC

O1O2O3O4 is an orthodiagonal quadrilateral.

2. If M,N,P,Q are tangent points of circles 1,2,3,4 to the circle with center O, then the quadrilateral MNPQ is inscribed in the circle with center O and

O, O1, M are colinear; O, O2, N are colinear; O, O3, P are colinear; O, O4, Q are colinear.

Best regards, Dinu

Dinu Teodorescu Joachim Domsta Ruslan Pozinkevych

I apologize to you all! The question was asked incorrectly—my mistake. Now everything is correct

@Dinu Teodorescu

You deserve A+A grade for presenting the answer before the question was posted. Cordial congratulations!

An accompanying problem forAll: Find algebraic relation between the positions of the centers and radia of the four incribed circles and the position of the tangent hords which would be equivalent to the existence of the big circle innerly tangent to all four circles?

right unless u want to change the assignment again Btw if it was a midschool exam the teacher would be fired :)

Dear Ruslan Pozinkevych

Can you share your way of obtaining pi/3, please?

sure but to do so the author must place the previous post again That's first Second the answer to the current question:546 png is right angle that is 90 degrees The solution way I leave to myself Im very confused when the question is withdrawn but the readers' answers remain It looks as though my answer pi/3 was given to the post 546 png.which is not the case When you ask me to explain how I got the answer pi/3 I must first get an old picture but it is not here due to the fact that the author withdrew the picture Now my apologies but these questions are not interesting to answer due to the reasons I mentioned Hope it helps Regards

PS. In the original picture the question was about value alfa of the measure of angle now named APD.

Ruslan Pozinkevych , Joachim Domsta Dear Ruslan Pozinkevych , if you can't provide a proof to show that your statements are valid, then the value of your statements represents exactly nothing!

To show minimal respect in this discussion, please offer to me and Professor Domsta some of your ideas to solve the initial problem, before Liudmyla modifies the statement! The png configuration is the same! Only the request is different!

Ruslan Pozinkevych Anyway, a picture for a geometrical configuration represents only a particular case! The motivation about solving the problem is important, not the picture!

I repeat, the picture is the same! You can see the validity of initial picture reading my initial comment, where you can see that the value of requested angle is 90o. See

"points O1, P, O3 are on a same line( are colinear), line which bisects the angle APD

points O2, P, O4 are on a same line( are colinear), line which bisects the angle APC

O1O2O3O4 is an orthodiagonal quadrilateral."

Dinu,

1. If there are no additional restrictions are imposed upon the positions, then the answer for value of value α of the angle APD is any number from the interval (0, π) and π/3 is a particular solution.

2. If the circles 1 and 4 have common tangent point on hord DC and if the circles 3 and 4 have common tangent point on hord AB then the answer for value of value α of the angle APD is

'any value from the interval (0,2 π/3)'.

It follows that

- π/3 is a particular solution

- π/3 is also the average of ALL solutions.

Ok I promised not to talk about this question Here is the answer though The question is in the picture It can be interpreted only one way: "What is the measure of the angle O1PO2"? The answer is: pi/2

Just in case APD angle is pi/3 For me this discussion is CLOSED

Dear Ruslan,

Honestly, I wrote those messages because I wanted to prove that you know what you're talking about! Your answer in the first message was aberrant anyway, because there are an infinity of values ​​of that angle for which the problem remains valid! I wanted to convince myself that you have elementary knowledge so that you can give an answer based on a true foundation. You didn't convince me by anything! I tried to find something in your works to convince me that you still have special skills in a certain field! I only managed to notice banal or elementary statements! And that doesn't give you the right to offend a teacher like Liudmyla, who, where she has her job, trains future math teachers, which is commendable! I don't expect any answer!

Joachim Domsta Dear Joachim, I understand all what you say, but, as you can see, my purpose was to show that mister RP has nothing to do with mathematics, elementary or advanced! Please visit his research section in his RG account, and please read his " math works"!

Regarding the problem, if r1, r2, r3, r4 are the radii of the four inscribed circles and R is the radius of circle with center O, then clearly the quantity r1+r2+r3+r4 is upper bounded by a quantity depending on R( or on R and the distance OP). Of course, it would be very interesting if we can find a maximum of quantity r1+r2+r3+r4 and the position of P for which we have equality with this maximum. I suspect we must have P is in O, but I don't have a proof!

And the maximum would be 4R(sqrt(2)-1), because in the situation P=O, if x=r1=r1=r1=r4, then 2 x^2=(R-x)^2 and it follows x=R(sqrt(2)-1).

But also we can study the problem of existence of a minimum ! Clearly, it's possible I'm wrong and 4R(sqrt(2)-1) be the minimum not the maximum....of course we need a proof for all suppositions I made!

Apologize ! I omited to say the most important thing! The position is P=O and the chords AB and CD are orthogonal!

I wouldn't refer to anything more than we see presented within this thread.

1. The first answer is wrong.

2. The answer is presented without proof despite its Esteemed Author has been asked for it.

3. The Esteemed Author of the first answer discredited the Author of the question for announcing a change of the problem under discussion WHICH IS NOT EVEN A MISTAKE, NOT TALKING ABOUT AN ERROR!

4. The Esteemed Author of the first answer started to treat the contributors as if they were not capable to see that his answers are inconsistent.

5. The Esteemed Author of the first answer tries to make the contributors responsible for his impoliteness.

simple does NOT have any resonate answer.

I would like to refute those outrageous accusations that one of the readers sent in my address which would be a defamation if it was not for a fact that I did make a mistake in evaluating the measure of the angle APD The answer is in definition and it's still a "circumscribed angle theorem" which is studied by the 8th Grade in Middle Schools But that is not the point The author who published the question changed it to which she admitted, quote:"I apologize to you all! The question was asked incorrectly—my mistake" So I gave the answer to the new question namely "Find the measure O1PO2 which I did just by looking into a schoolbook Now about myself I have been a teacher and a lecturer for over 20 years I worked at International schools Institutes etc. I did not deal with the school program quite a while and here is my initial mistake which I admit I made after quick assessment I will try not to respond to the questions just out of curiosity due to the comments posted by some readers After all Im under no obligation to explain why people make mistakes now and again Hope it helps Attached is the file to back up my reasoning Best Regards

There is a comment to the proof The angle O2PO1 is made by two bisecting lines of the supplementary angles DPA and APC and therefore is 90 degrees A+B=180→1/2A+1/2B=1/2(A+B)=90

Further proof (visual)

@Liudmyla Hetmanenko,

Dear Liudmyla,

I my opinion as of a mathematician your primary question was not asked incorrectly. It was a question for which the answer is of type "it depends . . . [here a bunch of admitted cases] ". Thus there is no reason for apology from your side. By doing this you have done simply kindly informed the readers that it was not the question you wanted to ask. What you have done is just an ADITIONAL question related to the same geometric circumstsnces. Incidentsly, this NEW question related to the OLD picture has obtained the correct answer by @Dinu Teodorescu in post #6, already before you formulated thst new question (as a side effect of considerations devoted to your primary question!) This makes the issues presented by you very interesting and nontrivial!

Unfortunately for this thread, a post evaluating your contribution as an error appeared. I wouldn't worry about this event since It cannot be read as a contribution to scientific or didactic activity. The more that also other posts of the author of answer #1 presented within this thread are logically inconsistent. Moreover, he dared to wrongly evaluate your contribution without ANY justified reasoning

Best regards,

Joachim Domsta

The question actually is not bad at all Regards

Any evaluation of correctness of a question may be related to its meaning, only. In particular if all terms in use are well defined. There are not and never have been such doubts about any of the TWO problems formulated within this thread. In both cases the problem is the value of the measure of PLAINLY DETERMINED angles: either DPA or O1 P O2. Correctness of these two questions is OBVIOUSLY independent of time. Thus it is a great confusion of the subject if one tries to suggest that the questions have changed their correctness.

1. Professional answer would contain a way in which pi/3 has been obtained by the author of post #1 or a statement that it is incorrect.

2. Instead of an erratum the author of post #1 replied (with first personal negative invective which has nothing to do with scientific dispute) [3 days ago]:

whereas changing the question of a problem during DISCUSSION is NOT an error and not even a mistake!

Opposite and adjacent angles made by two intersecting lines add up to 360 degrees With reference to the 546 picture the measure of a APD angle is a measure that could be any including the value of pi/3 Another story is with the supplementary angles APD and APC half of the measures of which will always add up to 90 So the answer to the question which was then removed a few days ago namely what is the measure of APD is "any" As you well know at schools such questions are considered to be ambiguous and hence are avoided Therefore the commentor said in a friendly manner that the teacher would face the consequences I admit that the commentor has overreacted but the question "What is the measure of the angle APD is such that it's difficult to define I also would not worry to much about why people make mistakes or admit to making them As Romans used to say ""Errare humanum est, perseverare autem diabolicum."

yes, IF we were in school THEN the answer pi/3 would be marked 'Failed', whereas it was presented in an 'overreacted'(?) way by words like this:

JoaD post #2 p.1: "PS. The answer by @Ruslan Pozinkevych is only one of possibilities. I think it is easy to see that with position of P in the center O with any angle between the hords we can inscribe the four circles."

PS: I am sorry for typos in name I've made in the original post.

No the answer pi/3 would be a failure but for different reasons Even if one was at the academy The reason for that is that it gives a specific answer to a question that has an infinite number of solutions Now turning back to the initial discussion about whether it is appropriate to ask about the measure of the angle APD For example APD? Yes it is And my mistake was to overreact However as far as the angle O1PO2 is concerned then there is no ambiguity in asking or answering the question Lastly I regret dragging myself into the discussion which was preposterous I retract all my comments and will not bother anyone who is involved in this discussion in any way based on a simple fact that "Errare humanum est, perseverare autem diabolicum."

All further references to my name related to my name are a mere provocation the main purpose of which is to discredit me and my work I apologize if I hurt somebody's feelings or offended smb unintentionally but my participation in this particular discussion and its members is finished I will not respond to any comments inside or outside of this blog Good afternoon

Dear Ruslan,

No one has anything with your level of knowledge in mathematics or the scientific level of articles written by you.

But when you make a mistake and refuse to accept that you were wrong, when you accuse someone unfairly, when you refuse to have a polite discussion with someone who asks you for an explanation.....then you have to understand and accept that there may be people who want to defend honesty and truth, and who want to punish you for the bad things you did!

However, as you said: "Errare humanum est, perseverare autem diabolicum." ! I positively appreciate your change of attitude! So, more attention, less haste, more knowledge in mathematics and good luck in the future!

All the best,

Dinu Teodorescu

@ Ruslan And please remember: in real life, those who do bad things will sooner or later get what they deserve! Which means that sooner or later, justice will prevail! Glory to Ukraine! Slava Ukraina! I am convinced your enemies will receive the punishment they deserve!

Dear Sir

This discussion serves no purpose I relieve myself from giving further comments due to the fact that there is very little constructivism regarding the question and it's solution I hope my answer is satisfactory enough if not consult other specialists and good luck to you too

For the sake of propriety

(see the picture attached)

The third equality in last answer is false even if the left hand side equals LN/FL.

Pls give substance to your comments I am posting my solution to the given problem and if there is a mistake in evaluation try your best to support your response with some evidence My claim is :angle O_1PO_2 measures=90 degrees, and the second statement (from the picture attached: FL/LN=angleAPD/angleDPB

Best of luck with your response

>

please do the best for presenting the substance of your proof.

Of course

Dear friends Pls do leave your valuable comments and try to evaluate the best answer Cheers

error #1: the presented proof is performed 'by a picture' which however use a specific configuration that the circles 1 and 4 are tangent.

I would rather remove, re-edit or add an erratum to any of my posts which happened to have content inconsistent with my will.

Joachim Domsta Dear Joachim, You don't need to talk to this idiot anymore! It's not worth wasting your time! He doesn't even understand that what he posts represents a series of writings that don't exist in mathematics..... ratio of measures of angles? It's stupidity without limits!

And it's not enough that he's stupid, the parrot is also naughty !

Joachim Domsta Please look at

Article ISOMORPHISM IN TERNARY MATHEMATICS MATRICES

to see how stupid this guy is!

I can't even believe there are journals that publish such imbecilities!

Dinu Teodorescu

My way has been suggested by Jonathan Doolin : to lead to self-contradiction if the claims are wrong and their author does not see this at once.

Best, Joachim Domsta

Joachim Domsta No chance! I think this guy is in a category which we know very well from thread about FLT !

Wow!

https://www.researchgate.net/post/Greatest_prime_numbers

Above text is brillant !!!

Also, here https://www.researchgate.net/post/On_3_59_as_valid2. The last post in page 1 seems to state that since (3+5)*0=9*0, we have (!) 3+5=9.

However, with reference to Liudmyla Hetmanenko's second figure (Screenshot_4.png), if I do not miss something, the circles with centers O1 and O2 are shown to be inscribed in the respective angles CPA and APD. This means that the two centres O1 and O2 lie on the bisectors of the angles CPA and APD, respectively, and since CPA+APD is equal to pi, the angle O1PO2 is equal to pi/2, as Ruslan Pozinkevych says.

First answer by R.P. was pi/3 without ANY explanation

Second complete answer on the subject even BEFORE change of the question (which happened without changing the pic), is due to Dinu Teodorescu who in post #6 on p.1 stated in the list of first propositions

Conversion degrees into numbers : multiply degrees (e.g angle measure) by pi/180 Make the respectful angle measure equal some number x,y…etc. Then substitute respectful numbers in the fraction For example x=60 times pi/180 and y=30 times pi/180 then you have a ratio x/y and off you go It can not be easier than that (in response to the comments about the ratio of the angles)

Refering to post #10 on p.4, the picture corresponds to particular case only when circles 1 and 4 are tangent (at point L). Under this assumption the following holds

FL/NL = tan( angle FNL) . .. . (in accordance with the text of the picture)

However, it is the same as

FL/NL = sin( 0.5 angle FPL) / sin (0.5 angle NPL),

where angle FPL+ angle NPL = pi. From properties of the sine function the ratio of the sine values equals the ratio of the angles only if they are equal. That means that the suggested by the text equality

FL/NL = angle FPL / angle NPL

is in general invalid, even under the assumption circles 1 and 4 are tangent (at point L). Not talking about any other cases.

Question. Is it possible that circles 1 and 4 are tangent (at point L) and the radii of circles 1 and 3 are NOT equal?

Nowhere in synthetic geometry books, in statements of theorems or problems, in proofs of theorems do ratios of angles appear ! It's not necessary ! There is no need for such a thing ! Therefore, writing something like this can be considered a proof of stupidity!

In the future, it would not be surprising if this RP offers us laconic texts containing ratio of triangles, ratio of quadrilaterals,...!

https://www.google.com/search?q=sine+law&sca_esv=455ef0c0e7ccb718&ei=XJ0PZ-mzArbdkPIP-orUwQ4&start=10&sa=N&sstk=AagrsuhmiQZDcDEn630Tx5G1tu1dtiYD76etDpsQTBgATNLqCE9_3V1RruqiU_6w2fRHH9A9yc6v_tAGaR4SXGm3w4YssriRgOa35A&ved=2ahUKEwjp0faW4JKJAxW2LkQIHXoFNegQ8tMDegQIBhAE&biw=1366&bih=641&dpr=1

To prove again his stupidity without limits, RP posts above a link containing facts about the sine theorem , which involves ratio of sines of angles, not ratio of angles, as he previous posted !!! - but it seems he can not understand that ratio of sines of two angles is not the same as ratio of measures of respective angles !

In a triangle ABC with lenghts a,b,c and angles A,B,C the relation

a/sinA=b/sinB=c/sinC holds( Sine Theorem), which implies , for example a/b=sinA/sinB.

It's about ratio of sines of angles, as you easy can see !

Post Scriptum: Ruslan, please stop infecting RG with your thinking of gibberish type.

Where did he say that ratio of sines of two angles is the same as ratio of measures of respective angles ? That is the first point

Second The commenter's English is hard to understand

Use 'as' after 'the same'. Don't use 'with' — it is not correct!

Finally The point in question was not my comments but the Maths question posted by the author Try to stick to the point

@Ruslan Pozinkevych asks with curiosity:

1. In post #10 on p.4 it is stated by Ruslan Pozinkevych that

FL/NL = angle FPL/angle NPL) [same as (0.5 angle FPL) / (0.5 angle NPL)].

2. In post #2 on p.6 I have proved that under conditions determined by the picture on #10 on p.4 we have

"FL/NL = sin( 0.5 angle FPL) / sin (0.5 angle NPL)".

3. Facts presented in points 1. and 2. are an evidence that Ruslan Pozinkevych stated that ratio of sines of two agles is the same as the ratio of measures of respective angles, which is not a mathematically correct . . . . . . q.e.d.

PS. I will never comment any next answer by Ruslan since they are either purposely aimed to disturb the scientific feature of this thread or present a deep lacks of fundametal methodology of mathematics. In either case such answers do NOT deserve devoting time for preparating suitable replies. Reason: they are NOT taken into account by Ruslan in any further discussion. For me continuing the discussion with his posts is a pure nonsense.

Joachim Domsta and Dinu Teodorescu from your comments it follows that you are not proposing a solution to the problem but make blatant remarks about what someone has said My point was expressed clearly and supported by the pictures I myself saw very little novelty in this discussion therefore offered a few solutions You may criticize them as you please but using profane and offensive remarks is another matter Here is my final explanation of how I wanted to answer the question "Who will attempt to solve the new problem and evaluate the author's idea?"

Check the attachment and please attempt to be professional if you can

Now all you have to do is to solve x/y=tan N and x+y=180 simultaneously I think this will be sufficient enough