From a Geometric definition, the medians of A ABC (lines from each vertex to the midpoint...

 

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From a Geometric definition, the medians of A ABC (lines from each vertex to the midpoint of the opposite side) divide the triangle into three smaller triangles of equal area. Point G, the intersection of the three medians, is the vertex common to all three smaller triangles and is the center of area. Some semi-involved Trigonometry will show why all the areas are equal. B E G (0,0) LL F A D C But alas, your task is to use Calculus to show the intersection of the medians is, in fact, (x,y)! To do this, you MUST use the diagram below for your proof, and follow the instructions. YA (b, c) (a,0) X a) Find the coordinates of the midpoints of all three sides. b) Find the equations for TWO of the median lines c) Find the x- and y-coordinate of the intersection of these two median lines. d) Find the equation for the third median line and verify that it also includes the point you found in part c. e) Find the necessary equations for the sides of the triangle, then use mighty Calculus to find the center of area (i.e. the sum of the moments in each direction divided by the area). From a Geometric definition, the medians of A ABC (lines from each vertex to the midpoint of the opposite side) divide the triangle into three smaller triangles of equal area. Point G, the intersection of the three medians, is the vertex common to all three smaller triangles and is the center of area. Some semi-involved Trigonometry will show why all the areas are equal. B E G (0,0) LL F A D C But alas, your task is to use Calculus to show the intersection of the medians is, in fact, (x,y)! To do this, you MUST use the diagram below for your proof, and follow the instructions. YA (b, c) (a,0) X a) Find the coordinates of the midpoints of all three sides. b) Find the equations for TWO of the median lines c) Find the x- and y-coordinate of the intersection of these two median lines. d) Find the equation for the third median line and verify that it also includes the point you found in part c. e) Find the necessary equations for the sides of the triangle, then use mighty Calculus to find the center of area (i.e. the sum of the moments in each direction divided by the area).