|
SPECIAL STRAIGHT LINES IN A TRIANGLE |
| Geometry | |
| 1.The medians of a triangle and the barycentre | |
| The median of a triangle is a straight line that joins one point (vertex) of a triangle to the midpoint of the opposite side. All three medians of a triangle coincide at a point called the barycentre. This point is the centre of gravity of a triangle, i.e. the point where we could tie a piece of string to the triangle and hang it horizontally. The following window shows us how the medians always coincide at one point regardless of the dimensions of the triangle. | |
| 1.- Check that the medians
always coincide at a point inside the triangle by moving points A, B
and C.
2.- In your exercise book and in the window construct a triangle whose sides are 3, 4 and 5 units long and another whose sides are 3, 5 and 7 units long and find their barycentres. 3.- Construct an equilateral triangle whose sides are 6 units long and draw in its medians.
|
|
| 2. The altitudes of a triangle and the orthocentre. | |
| The
altitude of a triangle is the perpendicular from one of its vertices
to the opposite side. The three altitudes of a triangle intersect at a
point called the orthocentre. The following Descartes window
will allow us to see how this is the case in any triangle we
construct. Depending on the type of triangle the orthocentre is
located inside the triangle, on one of its vertices or outside the
triangle.
|
|
|
4.- Check that the three altitudes of a triangle always intersect at a point by moving points A, B and C. 5.- Draw two triangles in your exercise book and in the window; one with sides 3, 4 and 5 units long and the other with sides 3, 5 and 7 units long. Find the orthocentre in each triangle 6.- Draw an equilateral triangle whose sides are 6 units long and draw its three altitudes. |
|
| 3. The perpendicular bisectors of a triangle and the circumcIRCLE. | |
| The perpendicular bisectors of a triangle are the perpendicular lines at the midpoint of each side. The perpendicular bisectors of the sides of any triangle intersect at a point called the circumcentre. The circumcentre is the centre of the circumcircle (the circle that passes through all the vertices of the triangle). | Use the window to check that the point where the three perpendicular bisectors of the sides of a triangle intersect at the centre of the circumcircle. To do so, drag the centre O, which passes through two of the vertices until it passes through all three. |
|
|
|
| 7.- Check that the perpendicular bisectors always intersect at a point by moving points A, B and C. Draw the following two triangles in your exercise book and in the window: one with sides 3, 4 and 5 and the other with sides 3, 5 and 7 and find the circumcentre of each one. Repeat the process for an equilateral triangle whose sides are 6 units long and draw in its perpendicular bisectors. | 8.- Draw a triangle with sides 5, 7 and 10 units long in your exercise book and draw in its perpendicular bisectors. Check that they intersect at a point inside the triangle and draw the circumcircle of the triangle. |
| 4. The angle bisectors of a triangle and the incIRCLE. | |
| The bisectors of the three angles of a triangle intersect at a point called the incentre. The incentre is the centre of the incircle (the inscribed circle drawn inside the triangle where each of the sides of the triangle is a tangent of the circle). | |
|
|
|
| 9.-
Move the red points to check that the three angle bisectors always
intersect at a point inside the triangle. 10.- Draw a triangle with sides 5, 7 and 10 in your exercise book and draw in its three angle bisectors. Check that these three lines intersect at a point inside the triangle. |
11.- Drag the centre O of this circle which forms tangents with two sides of the triangle until it also forms a tangent with the third side. Now you should be able to see that the centre coincides with the incentre. |
 |
 |
|||||
 |
| Miguel García Reyes | ||
 |
||
| Spanish Ministry of Education. Year 2001 | ||
Except where otherwise noted, this work is licensed under a Creative Common License