91Ó°ÊÓ

Construct a circle that passes thru a given point and is tangent to two intersecting lines.

\(A\) regular \(n\) -gon can be constructed with a ruler and a compass if and only if \(n\) is the product of a power of 2 and any number of distinct Fermat primes.

Given two line segments with lengths \(a\) and \(b,\) give a ruler-and-compass construction of a segments with lengths \(\frac{a^{2}}{b}\) and \(\sqrt{a \cdot b}\)

Assume that the initial configuration of geometric construction is given by the points \(A_{1}=(0,0), A_{2}=(1,0), A_{3}=\left(x_{3}, y_{3}\right), \ldots\) \(\ldots, A_{n}=\left(x_{n}, y_{n}\right) .\) Then a point \(X=(x, y)\) can be constructed using a compass-and-ruler construction if and only if both coordinates \(x\) and \(y\) can be expressed from the integer numbers and \(x_{3}, y_{3}, x_{4}, y_{4}, \ldots, x_{n}, y_{n}\) using the arithmetic operations "+","-",","/", and the square root " \(\sqrt{\text { " }}\)

Assume that the initial configuration of a geometric construction is given by the points \(A_{1}=(0,0), A_{2}=(1,0), A_{3}=\) \(=\left(x_{3}, y_{3}\right), \ldots, A_{n}=\left(x_{n}, y_{n}\right) .\) Then a point \(X=(x, y)\) can be constructed using a set-square construction if and only if both coordinates \(x\) and \(y\) can be expressed from the integer numbers and \(x_{3}, y_{3}, x_{4}, y_{4}, \ldots, x_{n}, y_{n}\) using the arithmetic operations "+", "-","", and"/" only.

Show that there is no ruler-only construction verifying that a given point is a midpoint of a given segment. In particular it is impossible to construct the midpoint only with a ruler.

There is no ruler-only construction verifying that \(a\) given point is the center of a given circle. In particular it is impossible to construct the center only with a ruler.