

A point of concurrency is the point where three or more lines intersect. 

We will start our investigation of the classical triangle centers with the centroid.
CENTROID  concurrent medians 
The median of a triangle is a segment joining any vertex to the midpoint of the opposite side. The medians of a triangle are concurrent (they intersect in one common point). The point of concurrency of the medians is called the centroid of the triangle. The medians of a triangle are always concurrent in the interior of the triangle.
The centroid divides the medians into a 2:1 ratio. The portion of the median nearest the vertex is twice as long as the portion connected to the midpoint of the triangle's side. For example, in ΔABC, shown above, if the length from C to the centroid is 10 units, then the distance from the centroid to P is 5 units.
Archimedes showed that the point where the medians are concurrent (the centroid) is the center of gravity of a triangular shape of uniform thickness and density.
If you cut a triangle out of cardboard and balance it on a pointed object, such as a pencil, the pencil will mark the location of the triangle's centroid (center of gravity or balance point). 

To locate the centroid through construction:
We have seen how to construct a median of a triangle. Simply construct the three medians of the triangle. The point where the medians intersect is the centroid.
Be sure to find the intersection of the medians (the red dot) and NOT the intersection of the segment bisectors used to locate the midpoints (the black dot).


Actually, finding the intersection of only 2 medians will find the centroid. Finding the third median, however, will ensure more accuracy of the find. 
FYI:
When working in the coordinate plane, the coordinates of the centroid of a triangle can be found by taking the average of the x coordinates of the three vertices, and the average of the y coordinates of the three vertices.
Add the xcoordinates and divide by 3.
Add the ycoordinates and divide by 3.
Centroid (1,2) 


PROVE: The medians of a triangle are concurrent (all intersect at one point).

It will be necessary to draw auxiliary lines to accomplish this proof. 


Plan of what needs to be done:
Draw a ray through A and F and intersecting at G.
Draw an auxiliary line through point B paralle to median . Label the intersection with the ray as point H.
Show that is a third median of Δ ABC by showing that G is the midpoint of . 

Outline of the Proof: The following things need to be accomplished to complete this proof.
•
Prove that ΔAFE is similar to ΔAHB.
•
Using the similar triangles, establish a proportion.
• Establish midsegment and get parallelogram.
• Use properties of parallelogram.
• Establish median.
Since all three medians pass through point F, the medians are concurrent. QED.
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