If the bisector of the vertical angle of the triangle bisects the base prove that the triangle is isosceles?
The law of sines says that the ratio of the common bisector line to the sine of the unbisected side angles must be the same. Therefore the two side angles must be equal. That makes the triangle isosceles.
To prove that the triangle is isosceles, we need to show that two of its sides are equal in length. Let's assume that we have a triangle ABC with the vertical angle at point A, and the bisector of the vertical angle bisecting the base BC at point D.
In order to prove that the triangle is isosceles, we can use the property of angles in a triangle, specifically the angle bisector theorem. According to this theorem, if a line bisects an angle of a triangle, it divides the opposite side into segments that are proportional to the lengths of the other two sides of the triangle.
In this case, we can apply the angle bisector theorem to angle A, which is the vertical angle. The vertical angle bisector line AD divides the opposite side BC into two segments, BD and DC.
Let's denote the lengths of the sides of the triangle as follows:
AB = a
AC = b
BC = c
BD = x
DC = c - x
According to the angle bisector theorem, we have the following proportion:
BD/DC = AB/AC
Substituting the values we defined earlier, we get:
x/(c - x) = a/b
Next, we can solve this proportion for x:
bx = ac - ax
bx + ax = ac
x(b + a) = ac
x = ac / (b + a)
Now, we need to prove that the triangle is isosceles, which means we need to show that two sides are equal. To do this, we can compare the lengths of the two sides of the triangle.
If AB = AC, then triangle ABC is isosceles. So, we need to compare a and b:
AB = a
AC = b
Now, let's derive the condition for a triangle to be isosceles from the equation we found earlier:
AB = AC
a = b
That means we need to prove that a = b.
Using the value of x we found earlier, we can rewrite the equation as:
x = ac / (b + a)
Substituting a = b, we get:
x = ac / (2a)
Simplifying further:
2ax = ac
2x = c
Now, substituting the value of x back into the equation, we have:
2(ac / (b + a)) = c
Multiplying both sides by (b + a), we get:
2ac = c(b + a)
Dividing both sides by c, we have:
2a = b + a
Simplifying, we get:
a = b
Thus, we have proven that if the bisector of the vertical angle of a triangle bisects the base, then the triangle is isosceles.