## Concave triangle

### Problem 587

A square is drawn around a circle as shown in the diagram below on the left.

We shall call the blue shaded region the L-section.

A line is drawn from the bottom left of the square to the top right as shown in the diagram on the right.

We shall call the orange shaded region a concave triangle.

It should be clear that the concave triangle occupies exactly half of the L-section.

Two circles are placed next to each other horizontally, a rectangle is drawn around both circles, and a line is drawn from the bottom left to the top right as shown in the diagram below.

This time the concave triangle occupies approximately 36.46% of the L-section.

If `n` circles are placed next to each other horizontally, a rectangle is drawn around the `n` circles, and a line is drawn from the bottom left to the top right, then it can be shown that the least value of `n` for which the concave triangle occupies less than 10% of the L-section is `n` = 15.

What is the least value of `n` for which the concave triangle occupies less than 0.1% of the L-section?