The length of the line between bisector's endings

Question

I've got isosceles triangle with two bisectors from the angles adjacent to it's base. My task is to find the length of the line between endings of bisectors (the point where it's connected with a side) in terms of a as a base of triangle and b as a side.

Answer 1

$A$ is the peak, and $B$ and $C$ are the base vertices. The angle bisectors hit the sides at points $D$ and $E$, and the line $DE$ is parallel to $BC$ by symmetry. Construct a segment from $E$, with endpoint $F$ on the line $AB$, such that $\angle BEC = \angle BEF$; then $\Delta BEC \cong \Delta BEF$ by angle-side-angle.

$$\angle BFE=\angle BCE=\angle ACB=\angle ABC=\angle ADE$$

Therefore, $\Delta DEF$ is isosceles, and $ED = EF = EC$.

Now just use triangle similarity $\Delta ABC \sim \Delta ADE$ for the ratio of sides

$$\frac{AC}{BC} = \frac{AE}{DE}$$

that is,

$$\frac{b}{a} = \frac{b-x}{x}$$ $$\frac1x = \frac1a + \frac1b$$ $$x = \frac{a \cdot b}{a + b}$$