Physics equations/Geometrical discussion of uniform circular motion

Using the figure we define the distance traveled by a particle during a brief time interval, $$\Delta t$$, and the (vector) change in velocity:

1    $$\Delta\ell= |\vec r_2 - \vec r_1|$$, and $$\Delta v= |\vec v_2 - \vec v_1|$$

2    $$\Delta\ell= v\Delta t$$ (rate times time equals distance).

3    $$\Delta\vec v =\vec a\Delta t$$ (definition of acceleration).

4    $$\Delta v = a\Delta t$$ (taking the absolute value of both sides).

5    $$\frac{\Delta v}{v} = \frac{\Delta\ell}{r}$$ (by similar triangles). Substituting (2) and (4) yields:

6    $$\frac{a\Delta t}{ v} =\frac{v\Delta t}{ r}$$, which leads to $$\frac{a}{v} = \frac{v}{r}$$, and therefore:

7    $$a=\frac{v^2}{r}$$