In triangle ABC, the angle bisectors AE and BD meet at O. Given AB = 12 cm, OA/OE = 3/2 and AD/DC = 6/7, find AC.

Step-by-step solution

1. 1

   Use the first bisector to get BC. $BD$ bisects angle $B$ and meets $AC$ at $D$, so the angle bisector theorem in triangle $ABC$ gives the ratio of the sides adjacent to B:

   $\frac{AD}{DC}=\frac{AB}{BC}\quad\Longrightarrow\quad \frac{6}{7}=\frac{12}{BC}\quad\Longrightarrow\quad BC=14\ \text{cm}$

   Getting the correspondence right matters: $AD$ is adjacent to $AB$, and $DC$ to $BC$.

2. 2

   Find the trick for the second condition. The ratio $\dfrac{OA}{OE}$ is measured along the segment $AE$, and $O$ is where $BD$ crosses it. Since $BD$ bisects angle $B$, the segment $BO$ is the bisector of angle $B$ inside triangle $ABE$ — because $A$, $O$, $E$ are collinear. Applying the bisector theorem to that smaller triangle:

   $\frac{OA}{OE}=\frac{BA}{BE}$

   This single observation is what makes the problem solvable with grade-school tools instead of coordinates.

3. 3

   Express BE in terms of the unknown AC. $AE$ bisects angle $A$, so in triangle $ABC$, $\dfrac{BE}{EC}=\dfrac{AB}{AC}$. With $BE+EC=BC=14$:

   $BE=BC\cdot\frac{AB}{AB+AC}=\frac{14\cdot12}{12+AC}=\frac{168}{12+AC}$

4. 4

   Substitute into the given ratio and solve.

   $\frac{OA}{OE}=\frac{AB}{BE}=\frac{12}{\dfrac{168}{12+AC}}=\frac{12\,(12+AC)}{168}=\frac{12+AC}{14}$

   Setting this equal to $\tfrac32$:

   $\frac{12+AC}{14}=\frac{3}{2}\quad\Longrightarrow\quad 12+AC=21\quad\Longrightarrow\quad AC=9\ \text{cm}$

5. 5

   Check the triangle is possible. With $AB=12$, $BC=14$, $AC=9$, every triangle inequality holds: $12+9=21>14$, $12+14>9$, $9+14>12$ ✓. So the data describe a genuine triangle rather than a degenerate one.

6. 6

   Verify both given ratios independently. With $AC=9$: $BE=\frac{14\cdot12}{21}=8$, so $\frac{AB}{BE}=\frac{12}{8}=\frac32$ ✓, matching $\frac{OA}{OE}$. And $\frac{AD}{DC}=\frac{AB}{BC}=\frac{12}{14}=\frac67$ ✓. Building the triangle in coordinates and intersecting the two bisectors explicitly gives $OA/OE=1.500000$ and $AD/DC=0.857143=\tfrac67$ ✓.