MHB Proof Quest: Non-Equilateral Triangle Enclosing Point Z

frusciante
Messages
3
Reaction score
0
I am struggling with this question, it would be easy enough if the triangle was equilateral but that is not necessarily the case.

Let (ha, hb, hc) be heights in the triangle ABC, and let Z be a point inside the triangle.

Further to this, consider the points P, Q, R on the sides AB, BC and AC, respectively. P, Q, R lie such that ZP is perpendicular to AB, ZQ is perpendicular to BC, and ZR is perpendicular to AC.

Show that ha/ZQ + hb/ZR + hc/ZP >= 9.

So far I have considered;
Area(ABC) = Area(AZC) + Area(AZB) + Area(BZC) = 0,5*(AC*ZR)+0,5*(AB*ZP)+0,5*(BC*ZQ)

But I don't know if that is any useful. Any comments?

Here is a sketched form of the triangle;

View attachment 5798

https://gyazo.com/7742bc1428ac5adf84033dfc2ec29564
 

Attachments

  • frusciante.jpg
    frusciante.jpg
    13.7 KB · Views: 108
Last edited by a moderator:
Mathematics news on Phys.org
Prove that $$\frac{ZP}{h_c}+\frac{ZQ}{h_a}+\frac{ZR}{h_b}=1$$. Then use the fact that $$(x+y+z)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)\ge9$$ (proved with the help of the harmonic mean).
 
Evgeny.Makarov said:
Prove that $$\frac{ZP}{h_c}+\frac{ZQ}{h_a}+\frac{ZR}{h_b}=1$$. Then use the fact that $$(x+y+z)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)\ge9$$ (proved with the help of the harmonic mean).

Thank you for your response, Evgeny. Indeed that first expression (=1) must hold, but I am having a hard time to prove that $$\frac{ZP}{h_c}+\frac{ZQ}{h_a}+\frac{ZR}{h_b}=1$$ since the two "types" of distances (ZP - h_c etc) aren't parallel given that the triangle is not equilateral. Any idea?
 
frusciante said:
I am having a hard time to prove that $$\frac{ZP}{h_c}+\frac{ZQ}{h_a}+\frac{ZR}{h_b}=1$$ since the two "types" of distances (ZP - h_c etc) aren't parallel given that the triangle is not equilateral. Any idea?
I am not sure what you mean by parallel types of distances. The altitude from $C$ and $ZP$ are parallel since they are both perpendicular to $AB$. I suggest multiplying both $h_c$ and $ZP$ by $AB$ and using your observation about the sum of areas.
 
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...
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...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.
Back
Top