Table Of Content

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 5 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 6 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 7 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 8 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 9