Complements to Classic Topics of Circles Geometry

2016·2.3 MB·English

by  Ion Pătrașcu

Ion Patrascu | Florentin Smarandache

Complements to Classic Topics
of Circles Geometry

Pons Editions
Brussels | 2016
Complements to Classic Topics of Circles Geometry

Ion Patrascu | Florentin Smarandache
Complements to Classic Topics
of Circles Geometry

1
Ion Patrascu, Florentin Smarandache

In the memory of the first author's father
Mihail Patrascu and the second author's
mother Maria (Marioara) Smarandache,
recently passed to eternity...

2
Complements to Classic Topics of Circles Geometry

Ion Patrascu | Florentin Smarandache

Complements to Classic Topics
of Circles Geometry

Pons Editions
Brussels | 2016
3
Ion Patrascu, Florentin Smarandache

© 2016 Ion Patrascu & Florentin Smarandache
All rights reserved. This book is protected by
copyright. No part of this book may be
reproduced in any form or by any means,
including photocopying or using any
information storage and retrieval system
without written permission from the
copyright owners.

ISBN 978-1-59973-465-1

4
Complements to Classic Topics of Circles Geometry
Contents
Introduction ....................................................... 15
Lemoine’s Circles ............................................... 17
1st Theorem. ........................................................... 17
Proof. ................................................................. 17
2nd Theorem. ......................................................... 19
Proof. ................................................................ 19
Remark. ............................................................ 21
References. ........................................................... 22
Lemoine’s Circles Radius Calculus ..................... 23
1st Theorem ........................................................... 23
Proof. ................................................................ 23
Consequences. ................................................... 25
1st Proposition. ...................................................... 25
Proof. ................................................................ 25
2nd Proposition. .................................................... 26
Proof. ................................................................ 27
Remarks. ........................................................... 28
3rd Proposition. ..................................................... 28
Proof. ................................................................ 28
Remark. ............................................................ 29
References. ........................................................... 29
Radical Axis of Lemoine’s Circles ........................ 31
1st Theorem. ........................................................... 31
2nd Theorem. .......................................................... 31
1st Remark. ......................................................... 31
1st Proposition. ...................................................... 32
Proof. ................................................................ 32
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Ion Patrascu, Florentin Smarandache
Comment. .......................................................... 34
2nd Remark. ....................................................... 34
2nd Proposition. .................................................... 34
References. ........................................................... 34
Generating Lemoine’s circles ............................. 35
1st Definition. ........................................................ 35
1st Proposition. ...................................................... 35
2nd Definition. ....................................................... 35
1st Theorem. .......................................................... 36
3rd Definition. ....................................................... 36
1st Lemma. ............................................................ 36
Proof. ................................................................ 36
Remark. ............................................................ 38
2nd Theorem. ......................................................... 38
Proof. ................................................................ 38
Further Remarks. .............................................. 40
References. ........................................................... 40
The Radical Circle of Ex-Inscribed Circles of a
Triangle ............................................................. 41
1st Theorem. .......................................................... 41
Proof. ................................................................ 41
2nd Theorem. ......................................................... 43
Proof. ................................................................ 43
Remark. ............................................................ 44
3rd Theorem. ......................................................... 44
Proof. ................................................................ 45
References. ........................................................... 48
The Polars of a Radical Center ........................... 49
1st Theorem. .......................................................... 49
Proof. ................................................................ 50
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Complements to Classic Topics of Circles Geometry
2nd Theorem. .......................................................... 51
Proof. ................................................................ 52
Remarks. ........................................................... 52
References. ........................................................... 53
Regarding the First Droz-Farny’s Circle ............. 55
1st Theorem. .......................................................... 55
Proof. ................................................................ 55
Remarks. ........................................................... 57
2nd Theorem. ......................................................... 57
Remark. ............................................................ 58
Definition. ............................................................ 58
Remark. ............................................................ 58
3rd Theorem. ......................................................... 58
Proof. ................................................................ 59
Remark. ............................................................ 60
4th Theorem. ......................................................... 60
Proof. ................................................................ 60
Reciprocally. ..................................................... 61
Remark. ............................................................ 62
References. ........................................................... 62
Regarding the Second Droz-Farny’s Circle.......... 63
1st Theorem. .......................................................... 63
Proof. ................................................................ 63
1st Proposition. ...................................................... 65
Proof. ................................................................ 65
Remarks. ........................................................... 66
2nd Theorem. ......................................................... 66
Proof. ................................................................ 67
Reciprocally. ..................................................... 67
Remarks. ........................................................... 68
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Ion Patrascu, Florentin Smarandache
2nd Proposition. .................................................... 68
Proof. ................................................................ 69
References. ........................................................... 70
Neuberg’s Orthogonal Circles ............................. 71
1st Definition. ......................................................... 71
2nd Definition. ....................................................... 72
3rd Definition. ....................................................... 72
1st Proposition. ...................................................... 72
Proof. ................................................................ 73
Consequence. .................................................... 74
2nd Proposition. .................................................... 74
Proof. ................................................................ 74
4th Definition. ....................................................... 75
3rd Proposition. ..................................................... 75
Proof. ................................................................ 75
Reciprocally. ..................................................... 76
4th Proposition. ..................................................... 76
Proof. ................................................................ 77
References. ........................................................... 78
Lucas’s Inner Circles .......................................... 79
1. Definition of the Lucas’s Inner Circles .......... 79
Definition. ............................................................ 80
2. Calculation of the Radius of the A-Lucas Inner
Circle .................................................................... 80
Note. ................................................................. 81
1st Remark. ........................................................ 81
3. Properties of the Lucas’s Inner Circles .......... 81
1st Theorem. .......................................................... 81
Proof. ................................................................ 81
2nd Definition. ....................................................... 83
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Complements to Classic Topics of Circles Geometry
Remark. ............................................................ 83
2nd Theorem. ......................................................... 84
3rd Definition. ....................................................... 84
3rd Theorem. ......................................................... 84
Proof. ................................................................ 85
Remarks. ........................................................... 86
4th Definition. ....................................................... 87
1st Property. .......................................................... 87
Proof. ................................................................ 87
2nd Property. ......................................................... 88
Proof. ................................................................ 88
References. ........................................................... 88
Theorems with Parallels Taken through a
Triangle’s Vertices and Constructions Performed
only with the Ruler ............................................ 89
1st Problem. ........................................................... 89
2nd Problem........................................................... 89
3rd Problem. .......................................................... 90
1st Lemma. ............................................................ 90
Proof. ................................................................ 91
Remark. ............................................................ 92
2nd Lemma. ........................................................... 92
Proof. ................................................................ 92
3rd Lemma. ........................................................... 93
Solution. ............................................................ 93
Construction’s Proof. ........................................ 93
Remark. ............................................................ 94
1st Theorem. ......................................................... 95
Proof. ................................................................ 96
2nd Theorem. ....................................................... 97
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