Triangle ▵ABC is provided.
Point X is along segment AC where |AX| : |XC| = 1 : x
and point Y is along segment BC where |BY| : |YC| = 1 : y.
Show all natural numbers x and y if the ratio of areas A▵ABC : A▵XYC = 2 : 1 is known.
(Solve this task without using Trigonometry.)
I'm not sure if triangles ▵ABC and ▵XYC are supposed to be similar or not.
I have an idea on how to solve this if they're similar.
It'd be nice to see someone post a solution that consider these triangles not similar.
Hint: Show that $ \frac{x}{1+x} \frac{y}{1+y} A\triangle ABC = A\triangle XYC$.
Hence, $ (x-1)(y-1) = 2 $ which only has solutions $(2,3), (3,2)$.