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Show that the marked region is half of an equilateral triangle

The angle \(\widehat{ABC} = 30^\circ\) because it is the complement of a right angle of an equilateral triangle.
The angle \(\widehat{ACB} = 60^\circ\) because the triangle \(ABC\) shares this angle with an equilateral triangle.
Hence the angle \(\widehat{BAC} = 180^\circ - 60^\circ-30^\circ = 90^\circ\), so the triangle \(ABC\) is a right triangle in \(A\).