Problem 7. Construct triangle $ABC$ from the following elements: a, $A, b+c$.

This ancient problem, according to al-Biruni [3], belongs to Menelaus (1st century AD), who formulated it as follows: inscribe a two-segment broken line of a given length in a given arc.
Al-Biruni's Solution. On the segment \( BC = a \) and angle \( A \), we construct a segment. We mark point \( D \) - the midpoint of the chord \( CB \) (Fig. 7). On \( DC \), as a diameter, we describe a circle and from point \( C \) with the compass set to \( CK = \frac{b+c}{2} \) (by Archimedes' theorem), we make a mark on it. Extending \( CK \) to intersect the segment, we get point \( A \). Triangle \( ABC \) is the desired one. In the process of investigation

![](https://cdn.mathpix.com/cropped/2024_05_21_c833ac06d3d78911cbb1g-013.jpg?height=577&width=540&top_left_y=908&top_left_x=1175)
we will find that the problem can have two symmetric solutions; one solution; and may have no solutions (when \( b+c < a \)).

![](https://cdn.mathpix.com/cropped/2024_05_21_c833ac06d3d78911cbb1g-013.jpg?height=682&width=668&top_left_y=1618&top_left_x=363)