# Isosceles triangle

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In geometry, an isosceles triangle is a triangle that has two sides of equal length. Sometimes it is specified as having two and only two sides of equal length, and sometimes as having at least two sides of equal length, the latter version thus including the equilateral triangle as a special case.

By the isosceles triangle theorem, the two angles opposite the equal sides are themselves equal, while if the third side is distinct then the third angle is distinct.

By the Steiner–Lehmus theorem, every triangle with two angle bisectors of equal lengths is isosceles.

## Symmetry

A triangle with exactly two equal sides has exactly one axis of symmetry, which goes through the vertex that is between the equal sides and also goes through the midpoint of the distinct side. Thus the axis of symmetry coincides with (1) the angle bisector of the distinct angle, (2) the median of the distinct side, (3) the altitude from the distinct side, and (4) the perpendicular bisector of the distinct side.

## Acute and obtuse

Whether the isosceles triangle is acute or obtuse depends on the distinct angle. The identical angles cannot be obtuse (greater than 90°) or right (equal to 90°) because their measures would sum to at least 180°, the total of all angles in any triangle. Since a triangle is obtuse if and only if one of its angles is obtuse, an isosceles triangle is obtuse if and only if its distinct angle is obtuse.

## Euler line

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 its three sides' perpendicular bisectors, which is the center of the circumcircle that passes through the three vertices). In a triangle with exactly two distinct sides, the Euler line coincides with the axis of symmetry. This can be seen as follows. 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.

If the distinct 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 distinct angle, and therefore the triangle, is obtuse, then the centroid still falls in the triangle's interior, but the orthocenter falls outside it (beyond the unique-angled vertex) and the circumcenter falls outside it (beyond the unique side).

In an isosceles triangle the incenter (the intersection of its angle bisectors, which is the center of the incircle which is internally tangent to the triangle's three sides) lies on the Euler line.

## Steiner inellipse

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 common side is longer than the distinct side then the Steiner inellipse's major axis coincides with the triangle's axis of symmetry; if the common side is shorter than the distinct side, then the ellipse's minor axis coincides with the triangle's axis of symmetry.

## Formulas

For an isosceles triangle with like sides of length b and distinct side of length a, the general triangle formulas for (1) the triangle-interior portion of the angle bisector of the distinct angle, (2) the median of the distinct side, (3) the altitude from the distinct side, and (4) the triangle-interior portion of the perpendicular bisector of the distinct side all simplify to $\tfrac{1}{2}\sqrt{4b^2-a^2}.$

Heron's formula for the area T of a triangle simplifies in the case of an isosceles triangle to

$T=\frac{a}{4}\sqrt{4b^2-a^2}.$

## Miscellaneous

If a cubic equation has two complex roots and one real root, then when these roots are plotted in the complex plane they are the vertices of an isosceles triangle whose axis of symmetry coincides with the horizontal (real) axis. This is because the complex roots are complex conjugates and hence are symmetric about the real axis.

Either diagonal of a rhombus divides it into two congruent isosceles triangles.