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.

Theorem

Let $△ A B C$ be a triangle in the Euclidean plane. Suppose squares $D A B E$ , $F B C G$ , and $H C A I$ are drawn on the outside of $△ A B C$ . Then the areas of the four triangles $△ A B C$ , $△ A I D$ , $△ B E F$ , and $△ C G H$ are equal.

Proofs

Proof by rotation

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

Proof by formula for area

Observe that $∠ C A B$ and $∠ D A I$ are supplementary. Therefore, we have

$\begin{matrix} [] [D A I] & = \frac{1}{2} | A D | | A I | \sin (∠ D A I) \\ = \frac{1}{2} | A B | | A C | \sin (∠ C A B) \\ = [A B C], \end{matrix}$

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

History

The theorem is named after David Cross, who discovered it around 2004. This configuration was also studied independently by Vecten^{[clarification needed]}, and consequently the theorem may also be called Vecten's theorem. However, the name "Vecten's theorem" is more commonly used for the theorem stating the existence of the Vecten points of a triangle.

References

1 2 3 "Cross Discovery". Cross's (Vecten's) theorem & generalizations to quadrilaterals.
↑ "Cross's Theorem". Cross's Theorem.
↑ Baker, Lydon; Harris, Ian (1 December 2004). "A Day to Remember". Mathematics Teaching. 189: 22.
↑ "ENCYCLOPEDIA OF TRIANGLE CENTERS". faculty.evansville.edu. Retrieved 2026-03-03.

[:0-1] 1 2 3 "Cross Discovery". Cross's (Vecten's) theorem & generalizations to quadrilaterals.

[2] "Cross's Theorem". Cross's Theorem.

[3] Baker, Lydon; Harris, Ian (1 December 2004). "A Day to Remember". Mathematics Teaching. 189: 22.

[4] "ENCYCLOPEDIA OF TRIANGLE CENTERS". faculty.evansville.edu. Retrieved 2026-03-03.