Answer:
Yes.
Explanation:
Yes.
You can prove it using SAS and CPCTC.
Start with an isosceles triangle with a horizontal base and the vertex angle above.
Draw the angle bisector at the vertex angle and extend it down until it intersects the base of the isosceles triangle.
Now you have two smaller triangles, right and left sides of the original triangle.
We now prove the two smaller triangles are congruent. The two congruent sides of the main triangle are congruent corresponding sides of the two smaller triangles. The bisected angle becomes two congruent angles in the two smaller triangles. Finally, the angle bisector is congruent to itself and is a side of both smaller triangles. By SAS, the two smaller triangles are congruent. Then, the two segments in the base of the isosceles triangle are congruent by CPCTC. That makes the point of intersection of the angle bisector and the base of the isosceles triangle the midpoint of the base.