ASSIGNMENT 8

BY

SHADRECK S CHITSONGA

ALTITUDES AND ORTHOCENTERS

 

Unlike the medians of the triangle that we discussed in assignment 6 we see that the orthocenter of a triangle can be found in several different positions depending on the triangle that one starts with. Refer to figure 1.

 

ORTHOCENTER

Figure 1

 

1. If we start with an equilateral triangle the orthocenter will be inside the triangle.

           2.  If the triangle is isosceles and not obtuse then the orthocenter is inside the triangle.

                3.  If the triangle is right angled then the orthocenter will be at the vertex where the right  

                     angle is.       

                4.  If the triangle is obtuse angled then the orthocenter is found outside the triangle.

 

                Note:  In the diagrams in figure 1 GSP was used to locate the orthocenter.

                           In the next discussion we will look at the actual constructions. Refer to figure 2.

                          H is the orthocenter for triangle ABC , the three red  circles are the circumcircles of    

                          triangles AHC,CHB  and AHB. The green circle is the circumcircle of the triangle 

                          ABC. All the constructions marks are left out just to illustrate  how one would do the

                          construction use compass and ruler. C1,C2,C3, and C4 are the circumcenters.

 

 

 

Figure 2

                

 We will leave out some of the constructions shown in figure 2, and concentrate on some observations that we can make from this construction. Refer to figure 3.

 

 

Figure 3

 

There are conjectures that we can make here, and that is all the four circles have the same radii,

        i.  The area of triangle formed by joining the circumcentres of the of the triangles

             HAC, HCB, and HAB is equal to the area of the original triangle.

       ii.   All the four circumcircles have the same radii.

       iii.   The orthocentres of the three triangles HBC, HAB, HAC coincide with the vertices of

              triangle ABC.

        iv. The hexagon formed by joining the centers of the circumcenters C1, C2,C3 and the

              vertices of the triangle ABC has an area equal to twice the area of triangle ABC.

       v. The sum of the areas of the triangles formed between each two vertices of the triangle

           ABC and each of the circumcentres is equal to the area of triangle ABC

 

Click HERE to see what happens when we move any of the three vertices to where H is.