[I'm living off my (meagre) savings and behind the firewall, so forgive me if this repeats your solution.]

If the line from the bottom bisects the other line, then the solution is straight forward -- and the area of the orange triangle, indeed, becomes irrelevant.

But it IS relevant to proving bisection.

Number the vertices 1, 3, 5, 7, 9, and 11, like on a clock. Connect v3 and v9 with a horizontal line. This is parallel to v11-v1, so the new small triangle in purple is similar to the orange one. We need to prov it is identical. Well, our new horizonal line bisects the line from the botom (v7), so A3's long side is indeed, 3 times A1's short side. The same apples to their heights, so the 3:1 ration checks out, and therefore their bases are equal. QED