http://jwilson.coe.uga.edu/EMAT6680Fa11/Antillanca/EMAT6680.gif


Rayen Antillanca. Assignment 8


The orthocenter

 

H is the orthocenter of ABC triangle. The orthocenter of a triangle is the point of concurrency of the three lines taken from a vertex of the triangle and perpendicular to the line of the opposite side.

The orthocenter is inside of the triangle if the triangle is acute; the orthocenter coincides with the vertex of the angle if the triangle is right, and the othocenter is outside of the triangle if the triangle is obtuse.

Acute triangle

Right triangle

Obtuse triangle

orth_acute.gif

orth_right.gif

orth_obtuse.gif

 

 

Given the ABC acute triangle

or1.gif

Where H is the orthocenter.

The orthocenter of triangle HBC is the vertex A, the Orthocenter of triangle HAB is the vertex C and the Orthocenter of triangle HAC is the vertex B. Why? As of these triangle are obtuse triangles, the orthocenter is outside of them and coincide with the opposite vertex.

I will show this fact in the figures below

The orthocenter of triangle HBC is vertex A

The orthocenter of triangle HAB is the vertex C

The orthocenter of triangle HAC is the vertex B

http://jwilson.coe.uga.edu/EMAT6680Fa11/Antillanca/Assignment8/OrHBC.gif

 

http://jwilson.coe.uga.edu/EMAT6680Fa11/Antillanca/Assignment8/OrHAB.gif

 

http://jwilson.coe.uga.edu/EMAT6680Fa11/Antillanca/Assignment8/OrHAC.gif

 

 

If I draw the circumcircle of triangles ABC, HBC, HAB, HAC, the result is the next image. Recall, the circumcircle is the circle which passes through all three vertices of a triangle

http://jwilson.coe.uga.edu/EMAT6680Fa11/Antillanca/Assignment8/or2.gif

 

 

When the orthocenter coincides with a vertex of the triangle ABC, all the triangles HBC, HAB, and HAC coincide with the triangle ABC. Similarly, all the circumcircles coincide with triangle ABCís circumcircles.

To see this, you can explore here.

 

 

The nine point circle

 

 

The nine point circle gets its name because it passes through nine significant points of a triangle. These points are:

         The midpoints of each side of the triangle

         The foot of each altitude

         The midpoint of the line segment that joints each vertex to the orthocenter.

 

The nine point circle of the acute triangle ABC is:

http://jwilson.coe.uga.edu/EMAT6680Fa11/Antillanca/Assignment8/9pABC.gif

 

 

 

The next images show the nine circle for triangle HBC, HAB, and HAC

Triangle HBC

Triangle HAB

Triangle HAC

http://jwilson.coe.uga.edu/EMAT6680Fa11/Antillanca/Assignment8/9pHBC.gif

 

http://jwilson.coe.uga.edu/EMAT6680Fa11/Antillanca/Assignment8/9pHAB.gif

 

http://jwilson.coe.uga.edu/EMAT6680Fa11/Antillanca/Assignment8/9pHAC.gif

As you can see, these triangles have the same nine point circle. With the next tool, you can draw the nine point circle.

 

 

Note: All figures and tools on this web page were made with The Geometerís Sketchpad

 

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