Inscribed Squares in Right-Angled Triangle/Side Lengths/Shared Right Angle

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Theorem

Let $a, b, c$ be the side lengths of a right-angled triangle, where $c$ is the length of the hypotenuse.

Then the side length $l$ of the inscribed square that shares a right angle with the right-angled triangle is given by:

$l = \dfrac {a b} {a + b}$


Proof

Inscribed-square-r.png

In the figure above, let $BC = a$ and $AC = b$.

Note that $DE \parallel CF$.

Therefore $\triangle BDE \sim \triangle BCA$ by Equiangular Triangles are Similar.

Thus:

\(\ds \frac {BD} {DE}\) \(=\) \(\ds \frac {BC} {CA}\) Definition of Similar Triangles
\(\ds \frac {a - l} l\) \(=\) \(\ds \frac a b\)
\(\ds b \paren {a - l}\) \(=\) \(\ds a l\)
\(\ds b a\) \(=\) \(\ds a l + b l\)
\(\ds l\) \(=\) \(\ds \frac {a b} {a + b}\)

$\blacksquare$


Also see