structural:expansion
Height

For any isosceles triangle, the following six line segments coincide:

Their common length is the height h {\displaystyle h} of the triangle.If the triangle has equal sides of length a {\displaystyle a} and base of length b {\displaystyle b},the general triangle formulas for the lengths of these segments all simplify to

This formula can also be derived from the Pythagorean theorem using the fact that the altitude bisects the base and partitions the isosceles triangle into two congruent right triangles.

The Euler line of any triangle goes through the triangle's orthocenter (the intersection of its three altitudes), its centroid (the intersection of its three medians), and its circumcenter (the intersection of the perpendicular bisectors of its three sides, which is also the center of the circumcircle that passes through the three vertices). In an isosceles triangle with exactly two equal sides, these three points are distinct, and (by symmetry) all lie on the symmetry axis of the triangle, from which it follows that the Euler line coincides with the axis of symmetry. The incenter of the triangle also lies on the Euler line, something that is not true for other triangles. If any two of an angle bisector, median, or altitude coincide in a given triangle, that triangle must be isosceles.