Procedures for geometric inversion

marcway shared this question 2 years ago

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.

supplementing:

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

see object c'' in attachment

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maybe this help.

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Hello Rami,

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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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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supplementing:

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

see object c'' in attachment

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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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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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.