SOLUTION: Conjecture: The angle bisector of the vertex angle of an isosceles triangle is also a median to the base.

The angle bisector divides it into two triangles which can
be proved congruent by SAS.  One pair of congruent sides are 
the congruent legs of the isosceles triangle.  The angles are 
congruent because an angle bisector divides the angle into two
congruent parts. The other pair of congruent sides is the 
angle bisector itself, as it is part of both triangles.  So 
you have SAS.

That makes the angle bisector also a bisector of the base 
because of corresponding parts of conruent triangles.

That is all that is needed to prove it is a median.

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It is also the perpendicular bisector of the base, which you may 
also be asked to prove later.
 
To do that, you prove additionally that the angles that the angle 
bisector make with the base are right angles.  First, they are 
congruent because of corresponding parts of congruent triangles.
Also they form a linear pair. Then congruent angles that form a 
linear pair are right angles.   

Edwin