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**heathrowjohnny** Let $\displaystyle \bold{r} $ be a point on a circle with diameter from $\displaystyle - \bold{a} $ to $\displaystyle \bold{a} $ (so that $\displaystyle |\bold{r}| = |\bold{a}| $). Draw the chord through $\displaystyle \bold{r} $ perpendicular to $\displaystyle \bold{a} $ (the length and direction of this chord is $\displaystyle 2 \bold{r}_{a \perp} $). Let $\displaystyle \bold{t} = \bold{r}_{a} $ be the point of intersection of this chord with the diameter. Prove that the square of half the length of the chord is the product of the distance from $\displaystyle -\bold{a} $ to $\displaystyle \bold{t} $ times the distance from $\displaystyle \bold{t} $ to $\displaystyle \bold{a} $: $\displaystyle |\bold{r}_{a \perp}|^{2} = (\bold{t} + \bold{a}) \cdot (\bold{a} - \bold{t}) $.