Triangle Problem?

Question

I have a bonus problem for calc2:

"Show that the medians (ie side bisectors) of a triangle intersect at the centroid of the triangle (geometric center)"

Im kinda confused about the question. You can find the medians easily...But in what direction do you extend them so they all interesect? You can make any 3 points anywhere interesect at the centroid if it is coming from a side. Im really confused.

Anyway, if someone could explain the problem a little better and then offer some TIPS that would be great.

You guys are awesome. The problem makes SO much more sense now. Im gonna try this on my own for a while and if I need any help Ill post a new question. Thanks a lot!

Answer 1

Bisect means to cut in half. so put a dot in the middle of each side of the triangle(on the line) and then draw a perpendicular from that point to the inside of the triangle. Do this for each side. The three lines that you drew will cross or intersect at the center of the triangle.

I hope that this helps you.

Answer 2

Adding to the mind map... There are these two circles (sorry, forgot the names) that are related to triangles.

From your description, you will get the centre of the circle inside the triangle just touching the three sides.

Reason: the line that joins from the middle of a side to the opposite vertex bisects the angle at the vertex. This line thus passes the center of a circle that just touches the other two sides, i.e. these two sides are tangents to the circle. Thus, the intersects of the three, or any two of such lines from middle of a side to the opposite vertex will be the center of the circle that has the sides of the triangle as its tangents.

To relate to centroid, each of the above lines divide the area of the triangle into equal halves. Treating the opposite side as the base, the line creates two triangles of the same height and equal base, which is half of the full base. Thus, the center of gravity lies on each of these lines and therefore is at the intersection of these lines.

Now, since this is also the centroid of the circle described above, it should mean that the centroid of the three parts of the triangle outside the circle is also at this point. Hmmmm... how to explain this?

An earlier answer mentioned about perpendicular bisectors of sides of a trangle. Such lines will pass through the center of circle that has the sides of the triangle as chords of it. Thus, the intersection of the bisectors is the center of the circle that has the three vertices of the triangle on its circumference.

Answer 3

If you draw a line from one vertex of a triangle to the midpoint of the opposite side, it will pass through the centroid. Do that for all three verticies / midpoints and they will intersect at the centroid. Showing why that is true is another story.

Answer 4

The problem refers to the three lines from the midpoint of each side to the opposite vertex. They'll intersect at the center of gravity.

Answer 5

it works best with an isosolis triangle mid point and geo- center are the same.