Prove that a string cant be outside a circle How can I prove that a chord can't be outside the circle itself.
Is there a way to prove that you can't draw a chord outside the circle.

Let $P_1 = (x_1,y_1)$ and $P_2 = (x_2,y_2)$ be points on the unit circle, and let $P=(x,y)$ be a point on the chord $P_1P_2$, not equal to $P_1$ or $P_2$. So we can write $P=tP_1 + (1-t)P_2$ for some $t \in (0,1)$.
So we have $x = tx_1 + (1-t)x_2$ and $y = ty_1 + (1-t)y_2$, giving
$$\begin{align}
x^2+y^2 & = t^2(x_1^2+y_1^2) + (1-t)^2(x_2^2+y_2^2) + 2t(1-t)(x_1x_2+y_1y_2) \\
& = t^2+(1-t)^2 + 2t(1-t)P_1.P_2\\
& < t^2+(1-t)^2 + 2t(1-t) \;(\mathrm{because}\;2t(1-t) > 0 \;\mathrm{and}\;P_1.P_2 < 1)\\
& = 1
\end{align}$$
