In an equilateral triangle, the bisector of the angle at the base is equal to b. Find the length of the side of a triangle if the angle at the base of this triangle is α.
Let in this triangle ABC the base is the AC side, the lateral sides are AB and BC, and the bisector of the angle at the base is AO.
Consider the AOC triangle.
In this triangle, ∠ОАС = α / 2, ∠ОСА = α, | AO | = b, therefore, by the sine theorem:
| OS | = b * sin (α / 2) / sinα = b / (2cos (α / 2)).
Consider triangle AOB.
In this triangle ∠ОАB = α / 2, ∠ABC = 180 – 2α, | AO | = b, therefore, by the sine theorem:
| OB | = b * sin (α / 2) / sin (180 – 2α) = bsin (α / 2) / sin (2α).
Therefore, | BC | = | OB | + | OС | = b / (2cos (α / 2)) + bsin (α / 2) / sin (2α).
Answer: b / (2cos (α / 2)) + bsin (α / 2) / sin (2α).
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