<s>[INST] An infinite sum of lengths. Let both the base and height of the following triangle have the length 6m. If the lines drawn inside the triangle bisect each right angle formed while proceeding. Then find the length of all these lines. The reason i could not do it is that i did not find any series representation of the lengths of the lines.   

 [/INST] First length element is

$$
\frac{1}{2} \sqrt{2} a
$$

Second length element is

$$
\frac{1}{2} a
$$

Now you repeat this for a smaller triangle with

$$
\frac{1}{2} a
$$

Then

$$
\Bigg( \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots \Bigg)
\Big( 1 + \sqrt{2} \Big) a
$$

Result

$$
= \Big( 1 + \sqrt{2} \Big) a
$$



$$
S = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots
$$

So

$$
2 S = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots
$$

Difference gives

$$
S = 2S - S = 1
$$

</s>