angle CDA = angle CDB and both are right angles, as CD is an altitude
from AB to C.
angle CAD = angle CBD as they are the angles opposite the equal sides
of an isosceles triangle.
angle ACD = angle DCB as these are the third angles in two triangles whose
other two angle are equal and so equal because the angle sum of any
triangle is two right angles.
Therefore as side CD is common to both triangle ACD and triangle BCD
these triangles are congruent by ASA.
Hence AD is congruent to DC as these are corresponding sides of congruent
triangles. So we have proven that D bisects AC.