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AD is a median of triangle ABC and BL and CM are perpendiculars

perpendiculars from B and C onto AD respectively... prove BL = CM



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See diagram.

ABC is a triangle.  AD is the median from A on to BC.  BD = DC.
Area of triangle ABD = 1/2 BD * AE  (altitude.
Area of triangle ADC = 1/2 DC * AE (altitude of triangle ADC also)

  As BD = DC,  the median divides the triangle into two equal parts (areas) ABD and ADC.

Now look at the area of triangle ABD with AD as the base. Area = 1/2 AD * BL
Area of triangle ADC with AD  as the base = 1/2 AD * CM
   Since the areas are equal,  CM = BL

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