The Centroid

1. The Centroid http://www.mathwithmccarthy.com/1/post/2012/01/balancing-act.html

2. Outline of Investigation • Investigate the intersection of the medians of a given triangle. • Six parts • Draw one example on graph paper. • Investigate using GeoGebra • Establish a conjecture • Prove the conjecture using areas of triangles • Prove the conjecture using coordinates • Prove the conjecture using dilations (later)

3. Using Quadrille Paper • Construct a triangle using most of one sheet of quadrille graph paper. It helps if the vertices occur at intersections of vertical and horizontal lines an even number of lines apart. • Indicate the midpoints of each of the sides. • Draw the medians – lines that connect the midpoints to the opposite vertex. • What do you notice?

4. Using GeoGebra • Construct points A,B,C and triangle ABC. • Construct midpoints D,E on sides AB, AC. • Construct medians CD, BE and their intersection point F. • Construct ray AF and its intersection (G) with side BC. • Compare lengths CG, BG. • Compare lengths AF, FG. • Formulate conjectures based on observations. • Find the point X that minimizes the sum of the squares of the distances to the vertices A, B, C.

5. A Proof—Using Areas • Given triangle ABC. • Construct TWO medians BE and CD. • Indicate F as the intersection of these TWO medians. • Construct ray AF, intersecting side BC at point G. • Let A1, … A6 denote the areas of the six triangles. • Denote p=CG and q=BG • Objective: prove that AG is the THIRD median, i.e., p=q

6. A Proof (cont’d) • A1=A6 and A2=A3 Reason:equal base lengths and heights. • A1+A2+A3=A6+A1+A2 Reason: each equals half the area of triangle ABC since E and D are midpoints. • Therefore, A3=A6 and A1=A2=A3=A6 Let T be this common value

7. A Proof (cont’d) • A4/A5=p/q and q(A4)=p(A5). REASON: Triangles CFG and BFG have the same altitudes • (A2+A3+A4)/(A1+A5+A6)=p/q => q(2T+A4)=p(2T+A5) REASON: Triangles AGC and AGB have the same altitudes • 2Tq + q(A4) = 2Tp + p(A5) =>p=q

8. A Proof Using Coordinates Conjecture: The medians of a triangle ABC intersect at a point that is 2/3 of the way from each vertex to the midpoint of the opposite side. Let the vertices of triangle ABC be represented as A: (a1,a2), B: (b1,b2), C: (c1,c2) • Determine the coordinates of the midpoint (G) of side BC. ANSWER: ( (b1+c1)/2, (b2+c2)/2 )

9. A Proof Using Coordinates (cont’d) 2. Determine the coordinates (x,y) of the point that is 2/3 of the way from A to G. ANSWER: x= (1/3)a1 + (2/3)(1/2)(b1+c1) (x,y) = ( (1/3)(a1+b1+c1), (1/3)(a2+b2+c2) ). 3. What is the corresponding result for the other medians? Explain. ANSWER: All that we need do is permute the symbols a,b,c. The result is the same point (x,y).

10. Interpretations & Applications • CENTROID is the point • for which the sum of the squares of the distances from the three vertices is as small as possible. • through which the best (least squares) linear fit to three data points is guaranteed to pass. • on which one can balance the triangle.

11. An Extension to 3D • Given 4 points A, B, C, D in 3D space. • Connect the points to form a tetrahedron (triangular pyramid). • Construct line segments from each vertex to the centroid of the opposite face. • These 4 segments intersect at a point that is 75% of the way from each vertex to the opposite face. • The coordinates of this point of concurrency is the average of the coordinates of the vertices, e.g., x=(1/4)(a1 + b1 + c1 + d1), and serve as the center of mass of a solid, uniform density tetrahedron.