# Line segment relationship in the circle

Given $AE=9,EB=8$; $DE:EC= 2:1$

Find $DE$ and $EC$

From the power of the point $E$, you have that $$EB\cdot AE= DE\cdot EC$$
Since you know that $AE=9$ and $AB=8$, then $DE\cdot EC=72$. Now, you know too that
$$\frac{DE}{EC}=2$$
So $DE=2EC$, and you have from the first equation that $DE\cdot EC= 2\cdot EC^2=72$, then $EC^2=36$, so $EC=6$ and $DE=12$.