 # Consider the triangle <mi mathvariant="normal">&#x0394;<!-- Δ --> A B C , which D is Ashley Fritz 2022-04-07 Answered
Consider the triangle $\mathrm{\Delta }ABC$ , which D is the midpoint of segment BC, and let the point G be defined such that $\left(GD\right)=\frac{1}{3}\left(AD\right)$ . Assuming that ${z}_{A},{z}_{B},{z}_{C}$ are the complex numbers representing the points (A, B, C):
a. Find the complex number ${z}_{G}$ that represents the point G
b. Show that $\left(CG\right)=\frac{2}{3}\left(CF\right)$ and that F is the midpoint of the segment (AB)
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Step 1
Let ${z}_{1},{z}_{2},{z}_{3}$ be the points A, B, C. Then it is clear that
$D=\frac{{z}_{2}+{z}_{3}}{2}$
The parametric equation of the line from D to A is
$\gamma \left(t\right)=\frac{{z}_{2}+{z}_{3}}{2}+t\left({z}_{1}-\frac{{z}_{2}+{z}_{3}}{2}\right)$
Therefore $G=\gamma \left(\frac{1}{3}\right)$ which you can simplify.
For the second part take the midpoint of AB and repeat the calculation. If you get exactly the same answer then you are done.

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