Question

X and Y are the points on side LN of triangle such that LX=XY =YNthrough X a line is drawn parallel to LM to meet MN at Z . prove that ar(LZY)=ar(MZYX).

Answer 1

given,
       LX=XY=YN
       XZ II LM
prove: ar (LZY)=ar(MZYX)
construction: join YZ,XM and LZ
proof:  ar(LZX)+(XZY)=ar(LZY)               --- (1)
           ar(MXZ)+ar(XZY)=ar(MZYX)        --- (2)
         
          ar(LZX)=ar(MXZ)       (both triangles are on the same base XZ and                                                             between same parallels LM and XZ)
          on adding equation (1) and (2) we get,
         ar(LZY)=ar(MZYX) 
          Hence proved