We want to construct a triangle and its medians. Since we have our library of GSP sketches then we can just find the sketch of the Centroid and play that sketch to get the triangle and its centroid. (Just hide the centroid we will not need it for this exercise.) To make things easy for discussion we will color each median a different color.
Our objective in this exercise is to construct a different triangle with the sides equal to the medians of the original triangle. My first thought was that I needed to use circles with radius equal to each of the medians. We need to construct a point p outside the original triangle and make p the center of our first circle. We can construct this circle by first picking point p and then highlighting the a median and then selecting the circle by center+ radius construction in the construct menu. We will keep our color code through out the assignment. Now we want to make a circle for each of the medians. However we cannot forget the purpose of the exercise: to make another triangle. Lets build the triangle as we go. Construct the next circle with the center on the first circle. We first construct a point on the red circle and then construct the circle just as before. Now where would we need to place the third center? Remember we want to make a triangle. I want the circles to intersect so that each side will be a radius of a different circle.
Now that we have our three intersections lets construct the triangle.
Click here to see a gsp script of the entire process.
Some interesting observations that I made concern the areas of the different triangles. I observed that the area of the new triangle is exactly 75% of the original triangle.
Click here and see for yourself.