Monday, February 27, 2012

Triangle Centers - II

Exploring Triangle Centres Using Geogebra
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
 The 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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