Whats the length of the angle bisector AM of the triangle ABC, if AB = c,
AC = b and ∠A = α
AM^2 = bc [1 − (a^2/(b + c)^2)]
see
geometry-math-journal.ro/pdf/Volume1-Issue1/ON%20THE%20STANDARD%20LENGTHS%20OF%20ANGLE%20BISECTORS%20AND%20THE%20ANGLE%20BISECTOR%20THEOREM.pdf
or
proofwiki.org/wiki/Length_of_Angle_Bisector
To find the length of the angle bisector AM of triangle ABC, we can use the Angle Bisector Theorem. The theorem states that the ratio of the lengths of the two segments formed by an angle bisector is equal to the ratio of the lengths of the opposite sides of the triangle.
Let's denote the length of the angle bisector AM as x. According to the Angle Bisector Theorem, we have:
AM/AB = AC/BC
Substituting the given values, we get:
x/c = b/BC
Now, we can solve for BC:
BC = (c * b) / x
Thus, the length of the angle bisector AM of triangle ABC is given by:
AM = BC - AB
= (c * b) / x - c
Therefore, the length of the angle bisector AM is (c * b - c * x) / x.
To find the length of the angle bisector AM of triangle ABC, we can use the Angle Bisector Theorem.
The Angle Bisector Theorem states that in a triangle, the angle bisector divides the opposite side into two segments that are proportional to the lengths of the other two sides.
In triangle ABC, let's label the length of the angle bisector AM as x. According to the Angle Bisector Theorem, we can set up the following proportion:
AM / AB = AC / BC
Substituting the given values, we have:
x / c = b / BC
To find the value of BC, we can use the Law of Cosines. The Law of Cosines states that in a triangle with sides a, b, and c, and angle C opposite side c, the following equation holds:
c^2 = a^2 + b^2 - 2ab * cos(C)
In our case, we know AB = c and ∠A = α. So, we have:
BC^2 = c^2 + b^2 - 2bc * cos(α)
Now, we can solve for BC by taking the square root of both sides:
BC = √(c^2 + b^2 - 2bc * cos(α))
Substituting this value of BC back into our proportion:
x / c = b / √(c^2 + b^2 - 2bc * cos(α))
To isolate x and solve for it, we can cross-multiply:
x * √(c^2 + b^2 - 2bc * cos(α)) = b * c
Finally, divide both sides by √(c^2 + b^2 - 2bc * cos(α)) to solve for x:
x = (b * c) / √(c^2 + b^2 - 2bc * cos(α))
Therefore, the length of the angle bisector AM of triangle ABC is (b * c) / √(c^2 + b^2 - 2bc * cos(α)).