Talk:Inscribed Squares in Right-Angled Triangle
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I haven't actually stopped to think about this, I'm just browsing latest changes overnight, but I wonder if there's a problem in this which would establish, for which specific dimensions of right triangle, the corner-positioned square or the side-positioned square is the greater? Someone must have asked the question. If I get a moment I may think about it myself, otherwise I'm just throwing it out there. --prime mover (talk) 09:10, 2 April 2022 (UTC)
- I had the same question when I wrote this page. The answer is the one in the corner is always larger:
| \(\ds l_h\) | \(=\) | \(\ds \frac {a b c} {a b + c^2}\) | Inscribed Squares in Right-Angled Triangle/Side Lengths/Side Lies on Hypotenuse | |||||||||||
| \(\ds \) | \(<\) | \(\ds \frac {a b c} {a c + b c}\) | Rearrangement Inequality: $c > a, b$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {a b} {a + b}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds l_r\) | Inscribed Squares in Right-Angled Triangle/Side Lengths/Shared Right Angle |
- The two-term rearrangement inequality can also be proven by $\paren {c - a} \paren {c - b} > 0 \leadstoandfrom c^2 + a b > a c + b c$.
- --RandomUndergrad (talk) 18:03, 2 April 2022 (UTC)
- Might be worth adding as a result in its own right. --prime mover (talk) 19:05, 2 April 2022 (UTC)