Triangles



Circumcenter, orthocenter, incenter and centroid

The circumcenter is equidistant from each vertex of the triangle.

The circumcenter is at the intersection of the perpendicular bisectors of the triangle's sides.

The circumcenter of a right triangle falls on the side opposite the right angle.


The incenter of a triangle is always inside it.

The incenter is where all of the bisectors of the angles of the triangle meet.

The incenter is equidistant from each side of the triangle.


The orthocenter of the triangle is where the three altitudes intersect at a single point

If the triangle is obtuse, then the orthocenter will be exterior to the triangle.

If the triangle is acute, then the orthocenter is located in the triangle's interior


The centroid is where the three medians of a triangle intersect.

The centroid is the triangle’s balance point, or center of gravity.

On each median, the distance from the vertex to the centroid is twice as long as the distance from the centroid to the midpoint of the side opposite the vertex.

The centroid is exactly 1/3 of the way from the midpoint of the side to the vertex of the triangle.

Given vertices (x,y) (a,b) and (m,n) the centroid coordinates would be ([x+a+m]/3, [y+b+n]/3)


PRACTICE PROBLEMS

Problem 1 :

Find the coordinates of the circumcenter of the triangle whose vertices are (2, -3), (8, -2) and (8, 6)


Problem 2 :

Find the coordinates of the circumcenter of the triangle whose vertices are (0,4) , (3, 6) and (-8, -2)


Problem 3 :

Find the coordinates of the orthocentre of the triangle whose vertices are (3, 1), (0, 4) and (-3, 1)


Problem 4: Find the co ordinates of the orthocentre of a triangle whose vertices are (2, -3) (8, -2) and (8, 6).

Problem 5: The coordinates of triangle ABC are A (0, 2), B (–2, 6), and C (4, 0). Find the coordinates of the orthocenter of this triangle.

Problem 6: The coordinates of triangle AND are A (0, 0), N (6, 0), and D (–2, 8). Find the coordinates of the orthocenter of this triangle.