# Isosceles triangle

Isosceles triangle

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 different then the third angle is different.

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

## Terminology

In an isosceles triangle that has exactly two equal sides, the equal sides are called legs and the third side is called the base. The angle included by the legs is called the vertex angle and the angles that have the base as one of their sides are called the base angles.[1]

Euclid defined an isosceles triangle as one having exactly two equal sides,[2] but modern treatments prefer to define them as having at least two equal sides, making equilateral triangles (with three equal sides) a special case of isosceles triangles.[3] In the equilateral triangle case, since all sides are equal, any side can be called the base, if needed, and the term leg is not generally used.

## Symmetry

A triangle with exactly two equal sides has exactly one axis of symmetry, which goes through the vertex angle and also goes through the midpoint of the base. Thus the axis of symmetry coincides with (1) the angle bisector of the vertex angle, (2) the median drawn to the base, (3) the altitude drawn from the vertex angle, and (4) the perpendicular bisector of the base.[4]

## Acute, right and obtuse

Whether the isosceles triangle is acute, right or obtuse depends on the vertex angle. In Euclidean geometry, the base 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 Euclidean triangle. Since a triangle is obtuse (resp. right) if and only if one of its angles is obtuse (resp. right), an isosceles triangle is obtuse, right or acute if and only if its vertex angle is respectively obtuse, right or acute.

## 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 an isoceles triangle with exactly two equal 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 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).

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.

## 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 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.

## Formulas

For an isosceles triangle with legs 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 $\tfrac{1}{2}\sqrt{4a^2-b^2}.$

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

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

This can be derived using the Pythagorean Theorem. The sum of the squares of half the base b and the height h is the square of either of the other two sides of length a.

$(\frac{b}{2})^2+h^2=a^2$

$h^2=a^2-\left(\frac{b}{2}\right)^2=a^2-\frac{b^2}{4}$

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

By substituting the height, the formula for the area of an isosceles triangle can be derived.

$T=\frac{1}{2}b\sqrt{a^2-\frac{b^2}{4}} =\frac{b}{4}\sqrt{4a^2-b^2}.$

## The isosceles triangle theorem

The theorem which states that the base angles of an isosceles triangle are equal appears as Proposition I.5 in Euclid.[5] This result has been called the pons asinorum (the bridge of asses). Some say that this is probably because of the diagram used by Euclid in his demonstration of the result. Others claim that the name stems from the fact that this is the first difficult result in Euclid, and acts to separate those who can understand Euclid's geometry from those who can't.[6]

## Partitioning into isosceles triangles

For any integer n ≥ 4, any triangle can be partitioned into n isosceles triangles.[7]

## 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.

## Fallacy of the isosceles triangle

A well known fallacy is the proof(?) of the statement that all triangles are isosceles. This argument has been attributed to Lewis Carroll,[8] but W.W. Rouse Ball claims priority in this matter.[9] The fallacy is rooted in Euclid's lack of recognition of the concept of betweeness and the resulting ambiguity of inside versus outside of figures.

## Notes

1. ^ Jacobs 1974, p. 144
2. ^ Heath 1956, p. 187, Definition 20
3. ^ Stahl 2003, p. 37
4. ^ Ostermann & Wanner 2012, p. 55, Exercise 7
5. ^ Heath 1956, p. 251
6. ^ Venema 2006, p. 89
7. ^ Lord, N. J. (June 1982), "Isosceles subdivisions of triangles", Mathematical Gazette 66: 136–137
8. ^ Robin Wilson (2008). Lewis Carroll in Numberland. Penguin Books. pp. 169–170. ISBN 978-0-14-101610-8.
9. ^ Ball, W.W. Rouse; Coxeter, H.S.M. (1987) [1892], Mathematical Recreations and Essays (13th ed.), Dover, p. 77 footnote, ISBN 0-486-25357-0

## References

• Heath, Thomas L. (1956), The Thirteen Books of Euclid's Elements 1 (2nd ed. [Facsimile. Original publication: Cambridge University Press, 1925] ed.), New York: Dover Publications, ISBN 0-486-60088-2
• Jacobs, Harold R. (1974), Geometry, W. H. Freeman and Co., ISBN 0-7167-0456-0
• Ostermann, Alexander; Wanner, Gerhard (2012), Geometry by Its History, Springer, ISBN 978-3-642-29162-3
• Stahl, Saul (2003), Geometry from Euclid to Knots, Prentice-Hall, ISBN 0-13-032927-4
• Venema, Gerard A. (2006), Foundations of Geometry, Prentice-Hall, ISBN 0-13-143700-3