Find area of triangle ABC given areas of sub-triangles The line p is parallel to the the side AB of triangle ABC and splits the sides AC and BC in points D and E, respectively. If the area of triangle ABD is m and the area of triangle AEC is n, find the area of ABC.


I think visually all you have to see is that $P_{ADE}=P_{BDE}$ (triangles with the same base between the same pair of parallel lines must have the same area.)
Let's set $AE\cap DB:=X$ for ease of notation
Since $P_{ADE}=P_{BDE}$, we have $P_{ADX}=P_{BEX}$ by subtracting $P_{DXE}$ from both sides of the original equation.
Notice by disection and associativity on multiplication:
$$P_{ABC}=P_{CDE}+P_{DXE}+P_{ADX}+P_{BEX}+P_{ABX}$$
$$=P_{ABX}+P_{ADX}+P_{CDE}+P_{DXE}+P_{BEX}$$
We further know that $P_{ABD}=P_{ABX}+P_{ADX}$ thus the equality becomes:
$$=P_{ABD}+P_{CDE}+P_{DXE}+P_{BEX}$$
Since $P_{ADX}=P_{BEX}$, we have
$$=P_{ABD}+P_{CDE}+P_{DXE}+P_{ADX}$$
Finally, $P_{AEC}=P_{CDE}+P_{DXE}+P_{ADX}$ due the disection and hence:
$$=P_{ABD}+P_{AEC}=m+n$$
This is a more explicit (though confusing due to all the letters) way of seeing it but it's really up to you. The others work great too! Hope that helps
