Isosceles and Equilateral Symmetry and Special Lines

Because isosceles triangles have two sides of equal length and two angles of equal length, some cevians in isosceles triangles have different properties than those from scalene triangles.

First, a isosceles triangle.

We will use this diagram for the below explanations.

1) The median from $A$ to $D$ is also the perpendicular bisector of $\overline{CB}$, the angle bisector of $\overline{AD}$, the altitude to $\overline{BC}$, and the symmedian from $A$.

2) The triangle centers $(O,H,I,G,N)$(Circumcenter, Orthocenter, Incenter, Centroid, Nine-point center respectively) lie on $\overline{AD}$, which is also the Euler line of the triangle.

For isosceles triangles, their line of symmetry is the altitude to the base of the triangle. For equilateral triangles, their lines are all of their altitudes.