## Task 4 - 180914

Given are two points $A_{0}$ and $A_{1}$. Their distance from each other is denoted by $a$.

Construct the vertices of a square with side length $3a$ using only a compass!
Circles with radius $a$ are drawn around $A_{0}$ and $A_{1}$. One of their intersection points is called $B_{1}$. Circles with radius $a$ are drawn around $A_{1}$ and $B_{1}$. Their intersection point different from $A_{0}$ is called $B_{2}$. Circles with radius $a$ are drawn around $A_{1}$ and $B_{2}$. Their intersection point different from $B_{1}$ is called $A_{2}$.

Since the triangles $A_{0} A_{1} B_{1}$, $B_{1} A_{1} B_{2}$ are equilateral, the angles $\angle A_{0} A_{1} B_{1}$, $\angle B_{1} A_{1} B_{2}$, and $\angle B_{2} A_{1} A_{2}$ each have a size of $60^{\circ}$ and together form a straight angle.

Thus, the points $A_{0}$, $A_{1}$, and $A_{2}$ lie on the same line, and $A_{0} A_{1} = A_{1} A_{2} = a$.

By continuing this process, a set of points $A_{n}$ can be constructed, all of which lie on the same line and any two adjacent points have a distance of $a$.

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1004.jpg?height=528&width=646&top_left_y=1164&top_left_x=705)

Using this method, a sequence of points $A_{0}, A_{1}, \ldots, A_{5}$ is constructed. Then $A_{0} A_{5} = 5a$, $A_{1} A_{4} = 3a$, and $A_{0} A_{4} = 4a$.

Now, a circle with radius $3a$ is drawn around $A_{4}$ and a circle with radius $5a$ around $A_{0}$. Both circles intersect, and one of the two intersection points is called $C$.

Since $(3a)^{2} + (4a)^{2} = (5a)^{2}$, by the converse of the Pythagorean theorem, the triangle determined by the points $A_{0}$, $A_{4}$, and $C$ is a right triangle, and $A_{4}$ is the vertex of the right angle.

Now, a circle with radius $3a$ is drawn around $C$ and a circle with the same radius around $A_{1}$. Their intersection point different from $A_{4}$ is called $D$. Since $A_{1} A_{4} = A_{4} C = C D = D A_{1} = 3a$ and the interior angle at $A_{4}$ is a right angle, $A_{1}$, $A_{4}$, $C$, and $D$ are the vertices of a square with side length $3a$.

Solutions of the I. Round 1978 taken from [5]

### 6.20.2 II. Round 1978, Class 9