Question

Triangle ABC and Triangle EBC are on same base BC. If AE
produced intersect BC at D prove ar(ABC)/ar(EBC)= AD/ED

Answer 1

See the diagram.

Draw perpendiculars AG and EH from A and E respectively on to BC.

ΔAGD and ΔEHD are similar triangles.  ∠G = ∠H = 90°. so AG || EH. ED|| AD.   GH||GD. 
=>  Hence, AD / ED = AG / EH    --- (1)

Area of ΔABC  = 1/2 * base * altitude 
  = 1/2 * BC * AG 
Area of ΔEBC   = 1/2 * BC * EH

Area(ΔABC) / Area(ΔEBC)
  = AG / EH
  = AD / ED     by (1)