Elements of triangles

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Medians and centroid

The line joining the midpoint of a side with the opposite vertex is called the median drawn to that side. Therefore, there will be three medians in every triangle. The three medians of a triangle meet at a point i.e. the three medians are concurrent. The point at which the medians meet is called centroid.

 Medians and centroid(G)

 

The centroid divides each of the medians in the ratio 2:1 with the greatest section being the one closer to the vertex.

The six triangles formed by the three medians of a triangle are equal in area and the area of each of these triangles is equal to one-sixth of the area of the original triangle.

In a right angled triangle, the length of the median drawn to the hypotenuse is equal to half the hypotenuse. This median is also the circumradius of the right angled triangle.

In an equilateral triangle, the lengths of all three medians are equal.

 

Altitude and orthocenter

The perpendicular line drawn from a vertex to the opposite side is the altitude of the triangle. There are three altitudes in a triangle which meet at a point. The point at which the altitudes meet is called the orthocenter of the triangle.

Altitudes and orthocenter(O)

 

In an acute angled triangle, the orthocenter lies inside the triangle. In a right angled triangle, the vertex where the right angle is formed is the orthocentre. In an obtuse angled triangle, the orthocenter lies outside the triangle.

 

Angular bisectors and incenter

Angular bisectors and incenter(I)

 

The bisectors of all the interior angles of a triangle meet at a point. This point is called the incenter of the triangle. The incenter is equidistant from all the sides of the triangle. Hence, a circle can be drawn tangential to all the sides of the triangle with incentre as the centre and the radius being the shortest distance from this centre to one of the sides. This circle is called the incircle of the triangle.

 

Perpendicular bisectors and circumcenter

Perpendicular bisectors and circumcenter(C1)

 

The perpendicular bisectors of all the sides of a triangle meet at a point. This point is called the circumcenter of the triangle. The circumcenter is equidistant from all the vertices of the triangle. Hence, a circle can be drawn passing through all the vertices of the triangle with circumcenter as the centre and the radius being the distance from this centre to one of the vertices. This circle is called the circumcircle of the triangle.


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