Medians AK and BM are drawn in triangles ABC, intersecting at point O.

Medians AK and BM are drawn in triangles ABC, intersecting at point O. Prove that the areas of triangles MOK and AOB are related as 1: 4

Consider triangles MOK and AOB:

KM connects the midpoints of the sides BC and AC, which means KM is the middle line of the triangle ABC.

By the property of the middle line, KM = 1 / 2AB, That is, KM refers to AB as 1/2.

By the property of the intersection of the medians of the triangle (the medians intersect in a 2: 1 ratio):

OM refers to BО as 1/2,

KO refers to AO as 1/2.

Consequently, the triangles MOK and AOB are similar (in the third attribute). The similarity coefficient is 1/2.

The areas of similar triangles are related as the square of the similarity coefficient:

S (MOK) / S (AOB) = (1/2) ² = 1/4.

Q.E.D.