<P> If the vertex angle is acute (so the isosceles triangle is an acute triangle), then the orthocenter, the centroid, and the circumcenter all fall inside the triangle . If the vertex angle, and therefore the triangle, is obtuse, then the centroid still falls in the triangle's interior, but the circumcenter falls outside it (beyond the base), and the orthocenter also falls outside the triangle (beyond the apex). </P> <P> In an isosceles triangle the incenter (the intersection of its angle bisectors, which is the center of the incircle, that is, the circle which is internally tangent to the triangle's three sides) lies on the Euler line . </P> <P> The Steiner inellipse of any triangle is the unique ellipse that is internally tangent to the triangle's three sides at their midpoints . In an isosceles triangle, if the legs are longer than the base then the Steiner inellipse's major axis coincides with the triangle's axis of symmetry; if the legs are shorter than the base, then the ellipse's minor axis coincides with the triangle's axis of symmetry . </P> <P> 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 1 2 4 a 2 − b 2 . (\ displaystyle (\ tfrac (1) (2)) (\ sqrt (4a ^ (2) - b ^ (2))).) </P>

The vertex angle of an isosceles triangle abc measures 30