Square with known diagonals

Statement

$ABCD$ is a square with $AB=BC=CD=DA=\ell$, $AP=2$ and congruent diagonals $CA\cong BD$. If $PA^2+PB^2+PC^2+PD^2=20$ and $\widehat{CAP}\cong\pi$, find $\ell$.

The square $ABCD$ with the external point $P$ joined to the four vertices.

Solution

Setting $CA=\sqrt{2}\,x$ (diagonal of the square of side $x$), the conditions translate into the system $\begin{cases} 4+2y^2+(2+x\sqrt{2})^2 = 20 \\ y^2 = \dfrac{x^2}{2}+4+4\sqrt{\dfrac{x^2}{2}}+\dfrac{x^2}{2} \end{cases}$ Extracting $y^2$ from the second equation and substituting it into the first, we obtain an equation in the single unknown $x$, from which the side $\ell$ of the square follows.

Links

Topics: Euclidean circle
Concepts: Inscribed quadrilateral
Skills: Synthetic geometry · Solving systems
Exercise type: Geometric problem