Draft for Information Only
ContentElementary Geometry
Elementary GeometryMiscellaneous Propositions976TheoremWhen three perpendiculars to the sides of a triangle 𝐴𝐵𝐶, intersecting them in the points 𝑎, 𝑏, 𝑐 respectively, are concurrent, the following relation is satisfied; and conversely, if the relation be satisfied, the perpendiculars are concurrent. 𝐴𝑏2−𝑏𝐶2+𝐶𝑎2−𝑎𝐵2+𝐵𝑐2−𝑐𝐴2=0ProofIf the perpendiculars meet in 𝑂, then 𝐴𝑏2−𝑏𝐶2=𝐴𝑂2−𝑂𝐶2, ⋯ (I.47).ExamplesBy the application of this theorem, the concurrence of the three perpendiculars is readily established in the following cases:
ProofIf 𝐴, 𝐵, 𝐶 and 𝐴′, 𝐵′, 𝐶′ are corresponding vertices of the triangle, join 𝐴𝐵′, 𝐴𝐶′, 𝐵𝐶′, 𝐵𝐴′, 𝐶𝐴′, 𝐶𝐵′, and apply the theorem in conjunction with (I.47)Sources and Referenceshttps://archive.org/details/synopsis-of-elementary-results-in-pure-and-applied-mathematics-pdfdrive©sideway ID: 210900026 Last Updated: 9/26/2021 Revision: 0 Ref: References
Latest Updated Links
|
Home 5 Business Management HBR 3 Information Recreation Hobbies 8 Culture Chinese 1097 English 339 Reference 79 Computer Hardware 249 Software Application 213 Digitization 32 Latex 52 Manim 205 KB 1 Numeric 19 Programming Web 289 Unicode 504 HTML 66 CSS 65 SVG 46 ASP.NET 270 OS 429 DeskTop 7 Python 72 Knowledge Mathematics Formulas 8 Algebra 84 Number Theory 206 Trigonometry 31 Geometry 34 Calculus 67 Engineering Tables 8 Mechanical Rigid Bodies Statics 92 Dynamics 37 Fluid 5 Control Acoustics 19 Natural Sciences Matter 1 Electric 27 Biology 1 |
Copyright © 2000-2024 Sideway . All rights reserved Disclaimers last modified on 06 September 2019