Triangles - Equilateral, Isosceles and Scalene

The triangle is a part of plane that is bounded by three line segments \(\overline{\rm AB}\), \(\overline{\rm BC}\) and \(\overline{\rm BC}\) that are called sides of a triangle as shown in the following figure.
The points of the triangle are usually labeled with letters A, B and C and these are vertices of triangles. As seen from previous figure the inner angles are usually labeled as \(\alpha, \beta\) and \(\gamma\). The sum of inner angles of the triangle is equal to 180° and is calculated using formula which can be written in the following form: $$\alpha + \beta + \gamma = 180°. $$ The triangles can be classified by the lengths of sides and by internal angles. Using the lengths of the sides classificiation there are equilateral triangle, isoscales triangle and scalene triangle. The equilateral trinagle has three sides of the same length. This type of triangle is a regular polygon with all angles measuring 60°. The example of equilateral triangle is shonw in the following figure.
The area of the equilateral triangle can be calculated from relation $$ A = \frac{\sqrt{3}}{4} a^2.$$ The perimeter of the equilateral triangle can be written as: $$ p = 3\cdot a. $$ The radius of the circumscribed circle i.e. the circle that passes through all vertices of equilateral triangle can be written as: $$R = \frac{a}{\sqrt{3}}.$$ The radius of the inscribed circle can be calculated as: $$ r = \frac{\sqrt{3}}{6}a.$$ The altitude (height from any side) can be calculated as: $$ h = \frac{\sqrt{3}}{2}a.$$ The isosceles triangle has two sides of equal length and it is shown in following figure. The isosceles triangle has two angles of the same measure which are opposite to the two sides of the same lenghts.
The height of the isosceles triangle can be calculated using formula derived from Pythagorean theorem as $$ h = \sqrt{b^2 - \frac{1}{4}a^2}.$$ The area is calculated using formula which can be written as: \begin{eqnarray} A &=& \frac{1}{2}ah\\ A &=& \frac{1}{2}a\sqrt{b^2 - \frac{1}{4}a^2}\\ A &=& \frac{1}{2}a^2 \sqrt{\frac{b^2}{a^2}-\frac{1}{4}}. \end{eqnarray} The scalene triangle has all tree triangle sides of different length. The area of scalene triangle can be calculated using formula: $$ P = \frac{a \cdot v_a}{2} = \frac{b\cdot v_b}{2} = \frac{c\cdot v_c }{2},$$ where \(a, b,\) and \(c\) are scalene triangle sides, and \(v_a, v_b,\) and \(v_c\) are heights on the corresponding sides of the triangle. The other formula for calculating area of the scalene triangle can be written as: $$ P = \sqrt{s(s-a)(s-b)(s-c)},$$ where parameter \(s\) can be calculated from $$ s = \frac{O}{2}=\frac{a+b+c}{2}.$$ The area of scalene triangle can be calculated with radius of incircle or circumcircle using fomrula: $$ P = rs = \frac{abc}{4R}$$. The area of scalene triangle can be calculated with the help of trigonometry as: $$ P = \frac{ab\sin\gamma}{2} = \frac{bc\sin\alpha}{2} = \frac{ac\sin\beta}{2},$$ $$ P = \frac{a^2 \sin\beta \sin \gamma}{2\sin\alpha} = \frac{b^2 \sin\alpha \sin\gamma}{2\sin\beta} = \frac{c^2\sin \alpha \sin \beta}{2\sin\gamma},$$ $$ P= 2R^2\sin\alpha\sin\beta\sin\gamma.$$
The triangles can be classified by the internal angles on right triangle, acute trinagle, and obtuse triangle. The right triangle is type of triangle in which one of the internal angles is right angle. The acute triangle is type of triangle in which one of the internal angles is acute angle. The obuste triangle is type of triangle in which one of the angles is obuste angle.
The altitude of a triangle is a straight line through a vertex and perpendicular to the opposite side. The opposite side is called the base of the altitude, and the foot of the altitude is the point where the altitude intersects the base. The altitude can alternatively be defined as the distance between the base and the vertex. In a triangle the total of three altitudes can be drawn that connects the vertex with opposite base. These three altitudes intersect in a single point called the orthocenter. It should be noted that the orthocenter of a triangle is inside the triangle if its acute triangle type (one of the angles is acute). The example of the altitudes and orthocenter is shonwin the following figure.
The line that goes through a vertex and the midpoint of the opposite side and divides the triangle into two equal areas is called median. The three medians intersect in a single point named centroid or geometric barycenter (denoted by G). The centroid of a rigid traingular object is also the center of mass.
The angle bisector of a triangle is a line that passes through a vertex which cuts the corresponding angle in half. In triangle the three angle bisectors intersect in a single point named incenter (denoted by letter I), whcih is the center of the trinagle's incircle. The incircle is the circle that lies inside the triangle and touches all three sides. The radius of the incircle is called inradius.
The stright line that is passing through the midpoint of the triangle side and is perpendicular to it is called perpendicular bisector. In triangle the three perpendicular bisectors meet in a single point named circumcenter (usually denoted by letter O). The circumcenter point is the center of circumcircle which is the cricle passing through all three vertices. The diameter of this circle is named circumdiameter while radius of such circle is called circumradius.