Procedures for geometric inversion

marcway shared this question 2 years ago
Answered

Hello everyone,


I would like help on the following question:


1- I have a construction with 2 circles: "c" and "d" - Circle c (Center O ', Radius 4) and Circle d (Center O, Radius 3).


2 - I am trying to reverse the Circle c, in relation to Circle d, geometrically.


3- I built the necessary points, but I do not know how to get the inversion that is a circle.


4- Geogebra can build it with the "Inversion" tool.


5 - Could someone explain step-by-step how to construct the circle of inversion through the use of geometric tools without using the Inversion tool?


Note 1: the construction is attached.


Note 2: Point B'' is the inversion of point B'.Thank you! Marcelo.

Best Answer
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supplementing:

how you receive in GGB, in all possible cases, 3 different Points of P'

see object c'' in attachment

Comments (8)

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1

maybe this help.

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Hello Rami,


Please forgive me the delay in answering your precious help.


I carefully analyzed its construction and understood "c '" as a locus.


It is perfect! Many thanks again for your explanations inside the Applet.


Thank you!


Marcelo.

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1

Hi marcway!

In your construction are some unnecessary points and lines.

For the construction look here: https://en.wikipedia.org/wiki/Inversive_geometry#Compass_and_straightedge_construction_(point_outside_circle)


To construct the inverse P' of a point P outside a circle Ø:

  • Draw the segment from O (center of circle Ø) to P.
  • Let M be the midpoint of OP.
  • Draw the circle c with center M going through P.
  • Let N and N' be the points where Ø and c intersect.
  • Draw segment NN'.
  • P' is where OP and NN' intersect.

We don't construct the points N and N' that way, we use the the "tangent-tool" to get the points C_1 and C_2. ;-)


See the corrected file - it's the same construction that you can see in rami's animated file.

I hope, the construction protocol will help you and you can activate the animation of the point B.


Greetings

mire2

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Hello Mire2,


Thank you very much for sending the file. I saw that my initial construction was wrong. Thank you for realizing and sending me the correction.


Thank you so much for the careful step by step that you sent me!


A careful tutorial helps us a lot to understand the geometric construction.


Thank you!


Marcelo.

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1

supplementing:

how you receive in GGB, in all possible cases, 3 different Points of P'

see object c'' in attachment

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1

Hello Rami,


Sensational ... in this construction I could easily see the 3 cases, by moving from the O' point.


Thank you so much for the class!


Marcelo.

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1

I think this version works in all (?) cases correct.

Also when radius d < radius c and d is full in c.

Pleas check.

2 cases:

P outside c: P --> tangent --> tangent-points c --> P'

P inside c: P --> tangent-points c --> tangent --> P' (mean reverse)

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1

Hello Rami,


Yes, I understood. I'm still analyzing the construction, so I can reproduce it in Geogebra.


Great!


Thank you so much for the precious help!


Marcelo.

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