ABCD is a trapezium with AB parallel to DC. a line parallel to AC intersects AB to X and BC to Y. prove that ar(ADX)=ar(ACY).

Join XC.

ADX and ACX lie on the same base AX and between the same parallels AX and DC.

ar (ADX)= ar (ACX)

ACY and ACX lie on the same base AC and between the same parallels AC and XY

  ar(ACY)=ar(ACX)

Since ar (ADX) = ar (ACX)

ar (ACY) = ar (ADX)