Two Proofs of One-Seventh Area Triangle
See https://www.geogebra.org/m/epzbrfxv for diagram
AB means length(AB) and ABC means area(ABC)
WLOG assume ABC has area 1. It'll be natural anyway.
Elementary proof:
Since, , and . By the same token . So .
Next,
Recall that if two triangles have the same altitude their area is the ratio of their bases.
This yields two sets of linear equations
Multiplying the second one by 2 and equating them we get
So since and ,
But , so
The same method can be applied to and , so they also have area . Now since , . So
A faster proof using Menelaus' theroem, since it is more powerful:
Which you would then use the last step of the above method to show that and then