{{Short description|Equality of triangles between three squares}} thumb|All of the red triangles have the same area.<!-- Which points in this image correspond to A, B, C, etc. in the statement of the theorem? Not labeling them is bad exposition. --> 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 <math>\triangle ABC</math> be a triangle in the Euclidean plane. Suppose squares <math>DABE</math>, <math>FBCG</math>, and <math>HCAI</math> are drawn on the outside of <math>\triangle ABC</math>. Then the areas of the four triangles <math>\triangle ABC</math>, <math>\triangle AID</math>, <math>\triangle BEF</math>, and <math>\triangle CGH</math> are equal.<ref name=":0">{{Cite web |title=Cross Discovery |url=https://dynamicmathematicslearning.com/crossdiscovery.html |website=Cross's (Vecten's) theorem & generalizations to quadrilaterals}}</ref><ref>{{Cite web |title=Cross's Theorem |url=https://www.wolframcloud.com/objects/demonstrations/CrosssTheorem-source.nb |website=Cross's Theorem}}</ref>
== Proofs == thumb|upright=0.8|After the outer triangles are rotated, each has an equal base and height to the central triangle
=== Proof by rotation === Rotate <math>\triangle AID</math> by a right angle, such that <math>D</math> coincides with <math>B</math>, and let this new triangle be <math>\triangle AI'B</math>. It is clear that <math>|AC|=|AI'|</math>, and that <math>C</math>, <math>A</math>, and <math>I</math> are collinear. Therefore, the areas of triangles <math>\triangle ABC</math> and <math>\triangle AI'B</math> are equal. Since <math>\triangle AI'B</math> is simply a rotation of <math>\triangle AID</math>, it follows that <math>\triangle ABC</math> and <math>\triangle AID</math> have the same area. Similar arguments prove the equality of all four areas. thumb|upright=0.8|Angles of the same color are supplementary.
=== Proof by formula for area === Observe that <math>\angle CAB</math> and <math>\angle DAI</math> are supplementary. Therefore, we have<blockquote><math>\begin{align}[] [DAI] &= \frac12|AD||AI|\sin(\angle DAI) \\ &=\frac12|AB||AC|\sin(\angle CAB) \\ &= [ABC], \end{align}</math></blockquote>as desired. Similar arguments show all four areas are equal.
== History == thumb|Construction of the outer Vecten point.
The theorem is named after David Cross, who discovered it around 2004.<ref name=":0" /><ref>{{Cite journal |last=Baker |first=Lydon |last2=Harris |first2=Ian |date=1 December 2004 |title=A Day to Remember |journal=Mathematics Teaching |volume=189 |pages=22}}</ref> This configuration was also studied independently by Vecten{{clarify|reason=Who is Vecten? When was Vecten active?|date=March 2026}}, and consequently the theorem may also be called Vecten's theorem.<ref name=":0" /> However, the name "Vecten's theorem" is more commonly used for the theorem stating the existence of the Vecten points of a triangle.<ref>{{Cite web |title=ENCYCLOPEDIA OF TRIANGLE CENTERS |url=https://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X485 |access-date=2026-03-03 |website=faculty.evansville.edu}}</ref>
== 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)
== References == {{Reflist}}
Category:Area Category:Angle Category:Articles containing proofs Category:Equations Category:Mathematical theorems Category:Euclidean plane geometry Category:History of geometry Category:Proof without words Category:Theorems_in_plane_geometry