Triangle BEC is similar to triangle ABC, point E belongs to AC, AE = 16cm, CE = 9 cm.

Triangle BEC is similar to triangle ABC, point E belongs to AC, AE = 16cm, CE = 9 cm. Angles ABC and BEC are obtuse. Find BC.

The angles of a triangle add up to 180 degrees, so
EBC + BCE + CEB = 180
ABC + BCA + CAB = 180
Since ABC = CEB, BCA = BCE then EBC = CAB
Apply the sine theorem
for a large triangle: AC / sin (ABC) = BC / sin (CAB)
for a small triangle: BC / sin (BEC) = EC / sin (EBC), use equal angles: BC / sin (ABC) = EC / sin (CAB)
from the equation for the big triangle we get:
sin (ABC) = sin (CAB) * AC / BC
substitute in the equations for the small triangle:
BC * BC / (sin (CAB) * AC) = EC / sin (CAB)
BC * BC = AC * EC = 16 * 9
BC = 12
Answer: 12 cm