Cross's theorem

All of the red triangles have the same area.

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

Proofs

After the outer triangles are rotated, each has an equal base and height to the central triangle

Proof by rotation

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

Angles of the same color are supplementary.

Proof by formula for area

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

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

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

History

Construction of the outer Vecten point.

The theorem is named after David Cross, who discovered it around 2004. This configuration was also studied independently by VectenTemplate:Clarify, 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.

See also

• Vecten points – Points formed by square centers
• Pythagorean theorem – Theorem relating areas of squares on the sides of a right-angled triangle.
• Bride's Chair – Illustration of Pythagorean theorem (and Cross's theorem without the flanks)