<abstract xmlns="http://www.w3.org/1999/xhtml">

For an arbitrary triangle <italic>ABC</italic> and an integer <italic>n</italic> we define points <italic>Dn</italic>, <italic>En</italic>, <italic>Fn</italic> on the sides <italic>BC</italic>, <italic>CA</italic>, <italic>AB</italic> respectively, in such a manner that
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_aupcsm-2016-0005_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mfrac><mrow><msup><mrow><mrow><mo stretchy="false">|</mo><mrow><mi mathvariant="italic">AC</mi></mrow><mo stretchy="false">|</mo></mrow></mrow><mi>n</mi></msup></mrow><mrow><msup><mrow><mrow><mo stretchy="false">|</mo><mrow><mi mathvariant="italic">AB</mi></mrow><mo stretchy="false">|</mo></mrow></mrow><mi>n</mi></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mrow><mo stretchy="false">|</mo><mrow><msub><mi mathvariant="italic">CD</mi><mi>n</mi></msub></mrow><mo stretchy="false">|</mo></mrow></mrow><mrow><mrow><mo stretchy="false">|</mo><mrow><msub><mi mathvariant="italic">BD</mi><mi>n</mi></msub></mrow><mo stretchy="false">|</mo></mrow></mrow></mfrac><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mfrac><mrow><msup><mrow><mrow><mo stretchy="false">|</mo><mrow><mi mathvariant="italic">AB</mi></mrow><mo stretchy="false">|</mo></mrow></mrow><mi>n</mi></msup></mrow><mrow><msup><mrow><mrow><mo stretchy="false">|</mo><mrow><mi mathvariant="italic">BC</mi></mrow><mo stretchy="false">|</mo></mrow></mrow><mi>n</mi></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mrow><mo stretchy="false">|</mo><mrow><msub><mi mathvariant="italic">AE</mi><mi>n</mi></msub></mrow><mo stretchy="false">|</mo></mrow></mrow><mrow><mrow><mo stretchy="false">|</mo><mrow><msub><mi mathvariant="italic">CE</mi><mi>n</mi></msub></mrow><mo stretchy="false">|</mo></mrow></mrow></mfrac><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mfrac><mrow><msup><mrow><mrow><mo stretchy="false">|</mo><mrow><mi mathvariant="italic">BC</mi></mrow><mo stretchy="false">|</mo></mrow></mrow><mi>n</mi></msup></mrow><mrow><msup><mrow><mrow><mo stretchy="false">|</mo><mrow><mi mathvariant="italic">AC</mi></mrow><mo stretchy="false">|</mo></mrow></mrow><mi>n</mi></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mrow><mo stretchy="false">|</mo><mrow><msub><mi mathvariant="italic">BF</mi><mi>n</mi></msub></mrow><mo stretchy="false">|</mo></mrow></mrow><mrow><mrow><mo stretchy="false">|</mo><mrow><msub><mi mathvariant="italic">AF</mi><mi>n</mi></msub></mrow><mo stretchy="false">|</mo></mrow></mrow></mfrac><mo>.</mo></mrow></mtd></mtr></mtable></math>
<tex-math>$$\matrix{{{{\left| {AC} \right|^n } \over {\left| {AB} \right|^n }} = {{\left| {CD_n } \right|} \over {\left| {BD_n } \right|}},} \hfill &amp; {{{\left| {AB} \right|^n } \over {\left| {BC} \right|^n }} = {{\left| {AE_n } \right|} \over {\left| {CE_n } \right|}},} \hfill &amp; {{{\left| {BC} \right|^n } \over {\left| {AC} \right|^n }} = {{\left| {BF_n } \right|} \over {\left| {AF_n } \right|}}.}} $$</tex-math>
</alternatives>
</disp-formula>
Cevians <italic>ADn</italic>, <italic>BEn</italic>, <italic>CFn</italic> are said to be the <italic>Maneeals of order n</italic>. In this paper we discuss some properties of the Maneeals and related objects.
</abstract>

For an arbitrary triangle ABC and an integer n we define points Dn, En, Fn on the sides BC, CA, AB respectively, in such a manner that



|AC|n|AB|n=|CDn||BDn|,|AB|n|BC|n=|AEn||CEn|,|BC|n|AC|n=|BFn||AFn|.
$$\matrix{{{{\left| {AC} \right|^n } \over {\left| {AB} \right|^n }} = {{\left| {CD_n } \right|} \over {\left| {BD_n } \right|}},} \hfill & {{{\left| {AB} \right|^n } \over {\left| {BC} \right|^n }} = {{\left| {AE_n } \right|} \over {\left| {CE_n } \right|}},} \hfill & {{{\left| {BC} \right|^n } \over {\left| {AC} \right|^n }} = {{\left| {BF_n } \right|} \over {\left| {AF_n } \right|}}.}} $$


Cevians ADn, BEn, CFn are said to be the Maneeals of order n. In this paper we discuss some properties of the Maneeals and related objects.

Several observations about Maneeals - a peculiar system of lines

Department of Mathematics, Kesava Reddy Educational Instutions, Machilipatnam, Kurnool, Andhra Pradesh,

Institute of Mathematics, Pedagogical University of Cracow, Podchorazych 2, 30-084 Kraków,

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.

{"article-title":"Several observations about Maneeals - a peculiar system of lines"}

For an arbitrary triangle ABC and an integer n we define points Dn, En, Fn on the sides BC, CA, AB respectively, in such a manner...

Several observations about Maneeals - a peculiar system of lines

Pubblicato online: 23 dic 2016

Pagine: 51 - 68

Ricevuto: 31 ago 2015

Accettato: 05 set 2016

DOI: https://doi.org/10.1515/aupcsm-2016-0005

Parole chiave
Maneeals, Maneeal’s Points, Maneeals triangle of order, Maneeal’s Pedal triangle of order, Cauchy-Schwarz inequality, Lemoine’s Pedal Triangle Theorem

© 2016 Naga Vijay Krishna Dasari et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.

Several observations about Maneeals - a peculiar system of lines

Pubblicato online: 23 dic 2016

Pagine: 51 - 68

Ricevuto: 31 ago 2015

Accettato: 05 set 2016

DOI: https://doi.org/10.1515/aupcsm-2016-0005

Parole chiaveManeeals, Maneeal’s Points, Maneeals triangle of order, Maneeal’s Pedal triangle of order, Cauchy-Schwarz inequality, Lemoine’s Pedal Triangle Theorem

© 2016 Naga Vijay Krishna Dasari et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.

Parole chiave
Maneeals, Maneeal’s Points, Maneeals triangle of order, Maneeal’s Pedal triangle of order, Cauchy-Schwarz inequality, Lemoine’s Pedal Triangle Theorem