Quadrilateral ABCD is a parallelogram. Line AK intersects line BC at point E, point K lies on the side of CD.

Quadrilateral ABCD is a parallelogram. Line AK intersects line BC at point E, point K lies on the side of CD. Find the ratio of the areas of triangles ABE and CKE, if AD = 12, CE = 4.

Let us prove that triangle ABE is similar to triangle KCE. Since ABCD is a parallelogram, then AB is parallel to CD, and therefore, by the main similarity lemma, the triangle ABE is similar to the triangle KCE (AB is parallel to the CK).
The areas of similar triangles are referred to as the squares of their respective sides or as the coefficient of similarity squared.
Relevant parties: AB and KS, BE and CE, AE and KE. BE refers to CE as well as 12 + 4 to 4 = 16/4 = 4. So the areas of these triangles are related as 4 ^ 2 = 16.
Answer: 16.