Wednesday, February 29, 2012
Monday, February 27, 2012
Triangle Centers - II
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
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
Saturday, February 25, 2012
Thursday, February 23, 2012
Tuesday, February 21, 2012
Monday, February 20, 2012
Saturday, February 18, 2012
Friday, February 17, 2012
Thursday, February 16, 2012
Tuesday, February 14, 2012
Monday, February 13, 2012
Triangle Centers - I
Visualizing Triangle Centers Using Geogebra
(Part-1)
- Introduction
Mark three non-collinear point P, Q and R on a paper. Join these pints in all possible ways. The segments are PQ, QR and RP. A simple close curve formed by these three segments is called a triangle.
A triangle is a 3-sided polygon. Every triangle has three sides and three angles, some of which may be have equal measurements. The sides of a triangle are given special names in the case of a right triangle, the side opposite to the right angle is called the hypotenuse and the other two sides being known as the legs. All triangles are convex. The portion of the plane enclosed by the triangle is called the triangle interior, while the remainder is the exterior.
Some basic facts about a triangle are
- Construction of a triangle is possible only when the sum of lengths of two sides is greater than the third side.
- The sum of angles in a triangle is 180°.
- In a triangle if any of the two sides are equal then angles opposite to equal sides are also equal. Such triangles are called isosceles triangles.
- In a triangle if three sides are equal , then the three interior angles are equal. Such triangles are called equilateral triangles.
- If a triangle has its three sides of unequal lengths then it is called a scalene triangle.
- Introduction to Geogebra
GeoGebra is an educational software for exploring and demonstrating Geometry and Algebra. It is an open source application and is freely available. It is capable of representing mathematical objects (at present 2-dimensional) algebraically and geometrically. For example, f(x) = x2 – 4x+2, a quadratic function, is represented by a parabola in graphical view and by an equation in algebraic window. A variable in Geogebra is represented by a slider. You can draw an object with the help of tools or by entering command in the input bar of GeoGebra window.
- Centers of Triangle
On every triangle there are points where special lines or circles intersect, and those points usually have very interesting geometrical properties. Such points are called triangle centers. Some examples of triangle centers are incenter, orthocenter, centroid, circumcenter, excenters, Feuerbach point, Fermat points, etc.
Concurrent Lines : Three lines are concurrent if there is a point P such that P lies on all three of the lines. The point P is called the point of concurrency. Three segments are concurrent if they have an interior point in common.
Two arbitrary lines will intersect in a point—unless the lines happen to be parallel. It is rare that three lines should have a point in common. One of the surprising and beautiful aspects of advanced Euclidean geometry is the fact that so many triples of lines determined by triangles are concurrent. Each of the triangle centers in this chapter is an example of that phenomenon.
- Centroid
A median of a triangle is a line segment that joins any vertex of the triangle to the mid point of the opposite side. There are three medians of a triangle. Three medians of the triangle meet at a point i.e. they are concurrent . This point of concurrency is knows as the centroid of the triangle.
Each median divides the triangle into two smaller triangles of equal areas.
One of the basic ideas known about the centroid is that it it divides the medians into a 2:1 ratio. The part of the median near to the vertex is always twice as long as the part near the midpoint of the side. If the coordinates of the triangle are known, then the coordinates of the centroid are the averages of the coordinates of the vertices. If we call the three vertices A(x1,y1) , B(x2,y2) and C(x3,y3) then the coordinates of the centroid are
In the following figure – 1 , points D , E and F are respectively the mid points of sides BC , CA and AB of triangle ABC. AD , BE and CF are the medians of triangle and G is its centroid.
Centroid of triangle always remains inside the triangle irrespective of its type (scalene , isosceles or equilateral)
Figure -1 Centroid (G) of a Triangle
- Circumcenter
A perpendicular bisector of a side of a triangle is a line which is perpendicular to the side and also passes through its mid point. There are three perpendicular bisectors of a triangle.
Figure – 2 Perpendicular Bisectors of a Triangle
Three perpendicular bisectors of a triangle meet at a point i.e. they are concurrent. This point is called circumcenter of the triangle. It is called circumcenter because it is the centre of the circle circumscribing the triangle. (a circle passing through the three vertices of the triangle). The distance of circumcenter from three vertices is equal and is the radius of the circumcircle.
For an acute triangle the circumcenter is inside the triangle , for obtuse triangle it lies outside and for a right triangle it lies at the mid point of the hypotenuse of the triangle.
Figure – 3 Circumcenter (C ) of an acute triangle
Figure – 4 Circumcenter (C ) of an obtuse triangle
Figure – 5 Circumcenter (C ) of a right triangle
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