# In an equilateral triangle, the bisector of the angle at the base is equal to b. Find the length of the side

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). One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.