Since as pointed out in the previous section the axis of symmetry coincides with an altitude, the intersection of the altitudes, which must lie on that altitude, must therefore lie on the axis of symmetry; since the axis coincides with a median, the intersection of the medians, which must lie on that median, must therefore lie on the axis of symmetry; and since the axis coincides with a perpendicular bisector, the intersection of the perpendicular bisectors, which must lie on that perpendicular bisector, must therefore lie on the axis of symmetry.
32.
Since as pointed out in the previous section the axis of symmetry coincides with an altitude, the intersection of the altitudes, which must lie on that altitude, must therefore lie on the axis of symmetry; since the axis coincides with a median, the intersection of the medians, which must lie on that median, must therefore lie on the axis of symmetry; and since the axis coincides with a perpendicular bisector, the intersection of the perpendicular bisectors, which must lie on that perpendicular bisector, must therefore lie on the axis of symmetry.
33.
For an isosceles triangle with equal sides of length " a " and base of length " b ", the general triangle formulas for ( 1 ) the length of the triangle-interior portion of the angle bisector of the vertex angle, ( 2 ) the length of the median drawn to the base, ( 3 ) length of the altitude drawn to the base, and ( 4 ) the length of the triangle-interior portion of the perpendicular bisector of the base all simplify to \ tfrac { 1 } { 2 } \ sqrt { 4a ^ 2-b ^ 2 }.
