In the triangle ABC, the bisectors BD and AE of the inner angles B and A intersect

In the triangle ABC, the bisectors BD and AE of the inner angles B and A intersect at point O. Calculate the length of the AC side if AB = 12, AO: OE = 3: 2 and AD: DC = 6: 7.

We will solve the problem based on the property of the bisector of the angle that it divides the opposite side into segments proportional to the adjacent sides.
We write down the ratio of the sides in the ABC triangle:
AB / BC = AD / DC = 6/7, 12 / BC = 6/7, → BC = 12 * 7/6 = 14.
We write down the ratio of the sides in the ABE triangle:
AB / BE = AO / OE = 3/2, 12 / BE = 3/2, → BE = 12 * 2/3 = 8.
EC = BC – BE = 14 – 8 = 6.
We write down the ratio of the sides in the ABC triangle:
AB / AC = BE / EC = 8/6, 12 / AC = 8/6, → AC = 12 * 6/8 = 9.
Answer: the AC side is 9.