SOLUTION: https://imgur.com/mlmAPzV In the diagram, angles marked are equal. BD = 4 cm and BC = 10 cm. Find the area, in cm^2, of triangle ABC.

In the Figure, you see three isosceles similar triangles ABC, BEC and CDE, listened from the largest to the smallest.


They all are isosceles, since they have congruent angles at their bases;
and they are similar, due to the same reason.


From isosceles triangle BEC, we have for its lateral sides  

   BD + DE = BC,  or  4 + DE = 10,  which gives  DE = 10-4 = 6.


Let x = EC.  Then from similarity triangles CDE and BEC we have this proportion

     =    for the ratio of their lateral sides to their base.


It gives x = EC = .



Let y = AB.  Then from similarity triangles ABC and BEC we have this proportion

     =    for the ratio of their lateral side to their base.


It gives y =  = .



Now we can find the altitude "h" of the triangle ABC.  It is  

    =  =  =  =  = ;

   h = .


Now the area of the triangle ABC is   = 59.5119 cm^2  (rounded).    ANSWER