**Visualizing Triangle Centers Using Geogebra**

**(Part-2)**

In my previous post http://mathematicsbhilai.blogspot.in/2012/02/triangle-centers-i.html
, I dicussed about Centroid
and Circumcenter of a triangle. In this post we
will discuss about Incenter and Orthocenter of a
triangle. Further we will discuss about Euler Line and Nine Point Circle .

**a.**

**Incenter**

An angle
bisector of a triangle is a line segment that bisects an angle of the triangle.

**Figure – 1 Angle Bisectors**

There are three angle bisectors of a
triangle.

Three angle
bisectors of a triangle meet at a point or they are concurrent. This point is
called incenter of the triangle. It is called the incenter because it is the centre of the circle inscribed (the largest circle that will
fit inside the triangle) in the triangle.

Centroid of triangle always
remains inside the triangle irrespective of its type (scalene
, isosceles or equilateral)

**Figure – 2 Incenter (I) of a triangle**

**b.**

**Orthocenter**

**T**he altitude of a triangle is a line which passes through a vertex of the triangle and is perpendicular to the opposite side. There are therefore three altitudes possible, one from each vertex.

**Figure – 3 Altitudes of a Triangle**

AF , BE and CF are three altitudes of triangle ABC.

The altitudes (perpendiculars from the vertices to the
opposite sides) of a triangle meet at a point i.e. they are concurrent . This point
is called orthocenter of the triangle.

For an acute
triangle the orthocenter is inside the triangle, for obtuse triangle it lies
outside and for a right triangle it lies at the vertex of the triangle where
right angle is formed.

**Figure – 4 Orthocenter (O) of an acute triangle**

There
is a very interesting fact
, If the orthocenter of triangle ABC is O, then the orthocenter
of triangle OBC is A, the orthocenter of
triangle OCA is B and the orthocenter of triangle OAB is C.

**Figure – 5 Orthocenter (O) of an obtuse triangle**

**Figure – 6 Orthocenter (O) of a right triangle**

**c.**

**Euler Line**

The
orthocenter O, the circumcenter C, and the centroid G of any triangle are collinear. Furthermore, G is
between O and C (unless the triangle is equilateral, in which case the three
points coincide) and OG = 2GC. The line
through O, C, and G is called the Euler line of the triangle.

**Figure – 7 Euler Line**

**d.**

**Nine Point Circle**

If DABC is any triangle, then the
midpoints of the sides of DABC, the feet of the altitudes of DABC, and the midpoints of the
segments joining the orthocenter of DABC to the three vertices of DABC all lie on single circle and this circle is called the
nine point circle.

Centre of nine point circle always
lies on the Euler line , and is the mid point of the
line segment joining orthocentre and circumcentre.

In an equilateral triangle
, the Orthocenter, centroid, and circumcenter conicide, so that
the Euler line has a length of 0. Further, the altitiudes
and medians are concurrent, so the 9-point circle now contains only 6
points.

**Figure –**8 Nine
Point Circle
(Scalene Triangle)

**Nine Point Circle**

**in an Isosceles Triangle**

In an
isosceles triangle the Euler line is collinear with the median from the vertex to the base. The
altitude and perpendicular bisectors to the base are the same, so the intersection of those
two lines with the base of the triangle is a coincident point. Thus our 9-point circle intersects 8 distinct
points. The obtuse isosceles triangle
also has 8
points in its 9-point circle.

**Figure –**9 Nine
Point Circle
(Isosceles Triangle)

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