## Task 3 - 040823

Given are the two adjacent angles $\alpha$ and $\beta$ with the vertex $A$ and point $D$ on the common side (see figure below).

a) Construct from this figure the triangle $A B C$ such that $\overline{A D}$ is a median!

b) Under what condition will the triangle $A B C$ be equilateral?

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0596.jpg?height=220&width=371&top_left_y=1112&top_left_x=1365)
a) One extends $\overline{A D}$ beyond $D$ by its own length and thus obtains point $E$. Then one constructs the parallelogram $A B E C$.

b) Exactly when $\alpha=\beta$ holds, $A B E C$ is a rhombus and thus $\triangle A B C$ is isosceles. The triangle $A B C$ is therefore equilateral if and only if $\alpha=\beta=30^{\circ}$.