MHB Right-Angled Triangle Inequality

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In a right triangle with sides a, b, and hypotenuse c, it is established that the inequality $$\frac{(c − a)(c − b)}{(c + a)(c + b)}\le 17 − 12\sqrt{2}$$ holds true. The discussion involves proving this inequality through various mathematical approaches. Participants explore different methods of manipulation and substitution to validate the inequality. The conversation emphasizes the importance of understanding the properties of right triangles in relation to the derived expression. Ultimately, the proof reinforces fundamental concepts in triangle inequalities.
anemone
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Show that if $a,\,b$ and $c$ are the lengths of the sides of a right triangle with hypotenuse $c$, then

$$\frac{(c − a)(c − b)}{(c + a)(c + b)}\le 17 − 12\sqrt{2}$$
 
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anemone said:
Show that if $a,\,b$ and $c$ are the lengths of the sides of a right triangle with hypotenuse $c$, then

$$\frac{(c − a)(c − b)}{(c + a)(c + b)}\le 17 − 12\sqrt{2}$$

From law of symmetry we shall have this shall have extremum at a = b giving $c= \sqrt{2}a$ and
$LHS = \frac{(c − a)(c − b)}{(c + a)(c + b)} = \frac{(\sqrt{2}-1)^2}{(\sqrt{2}+1)^2}= (\sqrt{2}-1)^4 = 17 - 12\sqrt{2}$
taking another point (a = 0, c=b) we get LHS = 0
hence $\frac{(c − a)(c − b)}{(c + a)(c + b)}\le 17 − 12\sqrt{2}$
 
kaliprasad said:
From law of symmetry we shall have this shall have extremum at a = b giving $c= \sqrt{2}a$ and
$LHS = \frac{(c − a)(c − b)}{(c + a)(c + b)} = \frac{(\sqrt{2}-1)^2}{(\sqrt{2}+1)^2}= (\sqrt{2}-1)^4 = 17 - 12\sqrt{2}$
taking another point (a = 0, c=b) we get LHS = 0
hence $\frac{(c − a)(c − b)}{(c + a)(c + b)}\le 17 − 12\sqrt{2}$
$a,b,\,\, and\,, c$ are the lengths of the sides of a right triangle
$a$ cannot be $"0"$
and more :
$c$ is hypotenuse ,we have $b<c$
 
Last edited:
Albert said:
$a,b,\,\, and\,, c$ are the lengths of the sides of a right triangle
$a$ cannot be $"0"$
and more :
$c$ is hypotenuse ,we have $b<c$

What you are telling is right. I took the limiting case
 
My solution:

We have $c^2−a^2=b^2 \implies c−a=\dfrac{b^2}{c+a}$.

By the similar token, we also have $c−b=\dfrac{a^2}{c+b}$, if we're going to replace these two into the original LHS of the inequality, we get:$$\begin{align*}\frac{(c − a)(c − b)}{(c + a)(c + b)}&=\frac{(b^2)(a^2)}{(c + a)^2(c + b)^2}\\&=\left(\frac{ab}{(c + a)(c + b)}\right)^2\\&=\left(\frac{ab}{c^2+c(a+b)+ab}\right)^2\\&=\left(\frac{1}{\dfrac{c^2}{ab}+\dfrac{c(a+b)}{ab}+1}\right)^2\\&\le \left(\frac{1}{\dfrac{c^2}{ab}+\dfrac{c(2\sqrt{ab})}{ab}+1}\right)^2\,\,\text{(by the AM-GM inequality)}\\& = \left(\frac{1}{\left(\dfrac{c}{\sqrt{ab}}\right)^2+\dfrac{2c}{\sqrt{ab}}+1}\right)^2\text{but from $c^2=a^2+b^2\ge 2ab$, we get $\dfrac{c}{\sqrt{ab}}\ge \sqrt{2}$}\\& \le \left(\frac{1}{\left(\sqrt{2}\right)^2+2\sqrt{2}+1}\right)^2\\&= \left(3-2\sqrt{2}\right)^2\\&= 17-12\sqrt{2}\,\,\,\text{Q.E.D.}\end{align*}$$
Equality occurs when $a=b$.
 
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