MHB Prove Triangle Inequality: AB/MZ + AC/ME + BC/MD ≥ 2t/r

  • Thread starter Thread starter solakis1
  • Start date Start date
  • Tags Tags
    Geometrical
AI Thread Summary
The discussion focuses on proving the inequality involving a triangle ABC and a point M inside it, where perpendiculars MZ, MD, and ME are drawn to the sides AB, BC, and AC, respectively. The goal is to demonstrate that the sum of the ratios of the triangle's sides to these perpendiculars is greater than or equal to twice the semi-perimeter divided by the inradius. The Cauchy-Schwarz inequality is suggested as a useful tool for the proof. Participants explore various approaches and mathematical principles to establish the validity of the inequality. The proof highlights the relationship between triangle geometry and inequalities, emphasizing the significance of the inscribed circle's radius.
solakis1
Messages
407
Reaction score
0
Given a triangle ABC and a point M inside the triangle ,draw perpendiculars MZ,MD,ME at the sides AB,BC,AC respectively. Then prove:$$\frac{AB}{MZ}+\frac{AC}{ME}+\frac{BC}{MD}\geq\frac{2t}{r}$$

Where t is half the perimeter of the triangle and r is the radius of the inscribed circle
 
Mathematics news on Phys.org
[sp]Here also the Cauchy-Schwarz inequality may be used for the solution of the problem[/sp]
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top