`A B C D` is a trapezium with `A B|\|D C` . A line parallel to `A C` intersects `A B ` at `X` and `B C` at `Ydot` Prove that `a r( A D X)=ar( A C Y)dot`

Explanation

Proof :
When two triangles lie on the same base and between the same parallels they must have equal area
`/_\ACX` and `/_\ ADX ......(1)` lie on the same base `AX` and between same parallels i.e `AX` abd `CD`
`/_\ACX` and `/_\ACY` lie on the same base `AC` and between same parallels i.e `AC` and `XY`
Then
`ar/_\ACX =ar/_\ACY........(2)`
hence
`ar(/_\ADX=ar(/_\ACY)`