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.