Answer:
10. Statement: Point P lies on line C'D'. Reason: The coordinates of P satisfy the equation of line C'D'.
Step-by-step explanation:
The equation for line C'D' in statement 8 appears to be in error. It should be ...
[tex]y=\dfrac{-s}{2t-r}x+\dfrac{2st}{2t-r}[/tex]
The coordinates of point P given in statement 9 are in error (they cannot be negative). They should be ...
P = (2/3(r+t), 2/3s) . . . . . without the minus signs
This point will satisfy the equation of line C'D'. Here is the algebra. (Notice we have multiplied each term on the right of the above equation by (-1/-1).
[tex]y=\dfrac{s}{r-2t}x-\dfrac{2st}{r-2t}\\\\\dfrac{2}{3}s=\dfrac{s}{r-2t}\cdot\dfrac{2}{3}(r+t)-\dfrac{2st}{r-2t}\qquad\text{substitute P for (x, y)}\\\\s(r-2t)=s(r+t)-3st\qquad\text{multiply by $\frac{3}{2}$(r-2t)}\\\\sr-2st=sr-2st\qquad\text{P satisfies the equation for line C'D'}[/tex]
So, step 10 should say, in effect, P lies on C'D'.
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Comment on the proof
IMO it is unfortunate that this proof has so many mistakes. Properly done, it is a nice demonstration that the medians are concurrent. It also demonstrates that the location of the centroid is 1/3 of the distance along the median toward the vertex.
