Hello, kapitanvicki!
A trapezium is divided into 4 triangles by its diagonals.
Prove the areas of the coloured triangles are equal. Code:
B C
* * * * * * *
*|:::* *:::|:*
*:|:::::::::*:::::::|:::::*
*::|:::::* O *:|:::::::*
*:::|:* | *:::::::*
*::* | | *:::*
*::* * * * * * * * * * * * * *
A E F D
We have trapezoid $\displaystyle ABCD$ with diagonals $\displaystyle AC$ and $\displaystyle BD$,
. . intersecting at $\displaystyle O.$
$\displaystyle \Delta ABD$ and $\displaystyle \Delta ACD$ have a common base, $\displaystyle AD$, and equal altitudes, $\displaystyle BE = CF.$
. . Hence: .$\displaystyle \text{area }\Delta ABD \:=\: \text{area }\Delta ACD$
They share $\displaystyle \Delta AOD.$
$\displaystyle \begin{array}{cccc}
\text{We have:} & \Delta ABD & = & \Delta ACD \\
\text{Subtract }\Delta AOD\!: & \text{-}\Delta AOD & = & \text{-}\Delta AOD \\
\text{Therefore:}& \Delta AOB & = & \Delta COD \end{array}$