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. Examples of isosceles triangles include the isosceles right triangle, the golden triangle, and the faces of bipyramids and certain Catalan solids.
The mathematical study of isosceles triangles dates back to ancient Egyptian mathematics and Babylonian mathematics. Isosceles triangles have been used as decoration from even earlier times, and appear frequently in architecture and design, for instance in the pediments and gables of buildings.
The two equal sides are called the legs and the third side is called the base of the triangle. The other dimensions of the triangle, such as its height, area, and perimeter, can be calculated by simple formulas from the lengths of the legs and base. Every isosceles triangle has an axis of symmetry along the perpendicular bisector of its base. The two angles opposite the legs are equal and are always acute, so the classification of the triangle as acute, right, or obtuse depends only on the angle between its two legs.
Formulas
Height
For any isosceles triangle, the following six line segments coincide:
the altitude, a line segment from the apex perpendicular to the base,
the angle bisector from the apex to the base,
the median from the apex to the midpoint of the base,
the perpendicular bisector of the base within the triangle,
the segment within the triangle of the unique axis of symmetry of the triangle, and
the segment within the triangle of the Euler line of the triangle.
Video: Height of Isosceles Triangle
Their common length is the height of the triangle. If the triangle has equal sides of length and base of length , 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 .....
In geometry two triangles, △ABC and △A′B′C′, are similar if and only if corresponding angles have the same measure: this implies that they are similar if and only if the lengths of corresponding sides are proportional.
Video: Similar Triangles
It can be shown that two triangles having congruent angles (equiangular triangles) are similar, that is, the corresponding sides can be proved to be proportional. This is known as the AAA similarity theorem. Due to this theorem, several authors simplify the definition of similar triangles to only require that the corresponding three angles are congruent.
There are several statements each of which is necessary and sufficient for two triangles to be similar:
The triangles have two congruent angles, which in Euclidean geometry implies that all their angles are congruent. That is:
If ∠BAC is equal in measure to ∠B′A′C′, and ∠ABC is equal in measure to ∠A′B′C′, then this implies that ∠ACB is equal in measure to ∠A′C′B′ and the triangles are similar.
All the corresponding sides have lengths in the same ratio:
AB/A′B′ = BC/B′C′ = AC/A′C′. This is equivalent to saying that one triangle (or its mirror image) is an enlargement of the other.
In geometry, an equilateral triangle is a triangle in which all three sides are equal.
In the familiar Euclidean geometry, equilateral triangles are also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. They are regular polygons, and can therefore also be referred to as regular triangles.
Video: What is an equilateral triangle
Principal properties
An equilateral triangle. It has equal sides (a=b=c), equal angles (), and equal altitudes (ha=hb=hc).
Denoting the common length of the sides of the equilateral triangle as a, we can determine using the Pythagorean theorem that:
The area is
The perimeter is
The radius of the circumscribed circle is
The radius of the inscribed circle is or
The geometric center of the triangle is the center of the circumscribed and inscribed circles
And the altitude (height) from any side is .
Denoting the radius of the circumscribed circle as R, we can determine using trigonometry that:
The area of the triangle is
Many of these quantities have simple relationships to the altitude ("h") of each vertex from the opposite side:
The area is
The height of the center from each side, or apothem, is
The radius of the circle circumscribing the three vertices is
The radius of the inscribed circle is
In an equilateral triangle, the altitudes, the angle bisectors, the ......
A right triangle (American English) or right-angled triangle (British English) is a triangle in which one angle is a right angle (that is, a 90-degree angle).
The relation between the sides and angles of a right triangle is the basis for trigonometry.
The side opposite the right angle is called the hypotenuse (side c in the figure). The sides adjacent to the right angle are called legs (or catheti, singular: cathetus). Side a may be identified as the side adjacent to angle B and opposed to (or opposite) angle A, while side b is the side adjacent to angle A and opposed to angle B.
If the lengths of all three sides of a right triangle are integers, the triangle is said to be a Pythagorean triangle and its side lengths are collectively known as a Pythagorean triple.
Video: Pythagorean theorem 3 | Right triangles and trigonometry
Principal properties
Area
As with any triangle, the area is equal to one half the base multiplied by the corresponding height. In a right triangle, if one leg is taken as the base then the other is height, so the area of a right triangle is one half the product of the two legs. As a formula the area T is
where a and b are the legs of the triangle.
Video: How to find the area of a right angled triangle
Altitudes
Altitude of a right triangle
If an altitude is drawn from the vertex with the right angle to the hypotenuse then the triangle is divided into two smaller triangles which are both similar to the original and therefore similar to each other. From this:
The altitude to the hypotenuse is the geometric mean (mean proportional) of the two segments of the hypotenuse.
Each leg of the triangle is the mean proportional of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg.
Video: How to Find the Altitude of a Right Triangle
In equations,
(this is sometimes known as the right triangle ......
In geometry, a hypotenuse (rarely: hypothenuse) is the longest side of a right-angled triangle, the side opposite of the right angle. The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides. For example, if one of the other sides has a length of 3 (when squared, 9) and the other has a length of 4 (when squared, 16), then their squares add up to 25. The length of the hypotenuse is the square root of 25, that is, 5.
Calculating the hypotenuse
A right-angled triangle and its hypotenuse, h, along with catheti (legs) c1 and c2.
The length of the hypotenuse is calculated using the square root function implied by the Pythagorean theorem. Using the common notation that the length of the two legs of the triangle (the sides perpendicular to each other) are a and b and that of the hypotenuse is c, we have
The Pythagorean theorem, and hence this length, can also be derived from the law of cosines by observing that the angle opposite the hypotenuse is 90° and noting that its cosine is 0:
Video: Pythagoras Theorem - Find Hypotenuse
Properties
In the figure, a is the hypotenuse and b and c are the catheti. The orthographic projection of b is m, and of c is n.
of the triangle. If the triangle has equal sides of length 
and base of length 
, the general triangle formulas for the lengths of these segments all simplify to
), and equal altitudes (ha=hb=hc).
or 
.
(this is sometimes known as the right triangle ......
