Fagnano's Theorem

If $P(x,y)$ and $P(x',y')$ are two points on an Ellipse

$${x^2\over a^2}+{y^2\over b^2}=1,$$ (1)

with Eccentric Angles $\phi$ and $\phi'$ such that

$$\tan\phi\tan\phi'={b\over a}$$ (2)

and $A=P(a,0)$ and $B=P(0,b)$. Then

$$\mathop{\rm arc}\nolimits BP+\mathop{\rm arc}\nolimits BP'={e^2 xx'\over a}.$$ (3)

This follows from the identity

$$E(u,k)+E(v,k)-E(k)=k^2\mathop{\rm sn}\nolimits (u,k)\mathop{\rm sn}\nolimits (v,k),$$ (4)

where $E(u,k)$ is an incomplete Elliptic Integral of the Second Kind, $E(k)$ is a complete Elliptic Integral of the Second Kind, and $\mathop{\rm sn}\nolimits (v,k)$ is a Jacobi Elliptic Function. If $P$ and $P'$ coincide, the point where they coincide is called Fagnano's Point.