Cross's theorem

In mathematics, specifically geometry, Cross's theorem, also known as Vecten's theorem, equates the area of a triangle to the area of each of the triangles formed by squares drawn along its sides.

Let ${\displaystyle \triangle ABC}$ be a triangle in the Euclidean plane. Suppose squares ${\displaystyle DABE}$, ${\displaystyle FBCG}$, and ${\displaystyle HCAI}$ are drawn on the outside of ${\displaystyle \triangle ABC}$. Then the areas of the four triangles ${\displaystyle \triangle ABC}$, ${\displaystyle \triangle AID}$, ${\displaystyle \triangle BEF}$, and ${\displaystyle \triangle CGH}$ are equal.¹²

Rotate ${\displaystyle \triangle AID}$ by a right angle, such that ${\displaystyle D}$ coincides with ${\displaystyle B}$, and let this new triangle be ${\displaystyle \triangle AI'B}$. It is clear that ${\displaystyle |AC|=|AI'|}$, and that ${\displaystyle C}$, ${\displaystyle A}$, and ${\displaystyle I}$ are collinear. Therefore, the areas of triangles ${\displaystyle \triangle ABC}$ and ${\displaystyle \triangle AI'B}$ are equal. Since ${\displaystyle \triangle AI'B}$ is simply a rotation of ${\displaystyle \triangle AID}$, it follows that ${\displaystyle \triangle ABC}$ and ${\displaystyle \triangle AID}$ have the same area. Similar arguments prove the equality of all four areas.

Observe that ${\displaystyle \angle CAB}$ and ${\displaystyle \angle DAI}$ are supplementary. Therefore, we have

${\displaystyle {\begin{aligned}[][DAI]&={\frac {1}{2}}|AD||AI|\sin(\angle DAI)\\&={\frac {1}{2}}|AB||AC|\sin(\angle CAB)\\&=[ABC],\end{aligned}}}$

as desired. Similar arguments show all four areas are equal.

The theorem is named after David Cross, who discovered it around 2004.¹³ This configuration was also studied independently by Vecten, and consequently the theorem may also be called Vecten's theorem.¹ However, the name "Vecten's theorem" is more commonly used for the theorem stating the existence of the Vecten points of a triangle.⁴