Understanding the Condition

The problem involves a quadrilateral ABCD where the diagonal AC bisects both angle A (\(\angle DAB\)) and angle C (\(\angle BCD\)). This implies:

• \(\angle BAC = \angle DAC\)
• \(\angle BCA = \angle DCA\)

Let's denote \(\angle BAC = \angle DAC = \alpha\) and \(\angle BCA = \angle DCA = \beta\).

Applying the Sine Rule

Using the Sine Rule in the triangles formed by the diagonal AC:

• In \(\triangle ABC\): \(\frac{AB}{\sin(\angle BCA)} = \frac{BC}{\sin(\angle BAC)}\) which becomes \(\frac{AB}{\sin \beta} = \frac{BC}{\sin \alpha}\).
• In \(\triangle ADC\): \(\frac{AD}{\sin(\angle DCA)} = \frac{DC}{\sin(\angle DAC)}\) which becomes \(\frac{AD}{\sin \beta} = \frac{DC}{\sin \alpha}\).

From these, we derive the ratios:

• From \(\triangle ABC\): \(\frac{AB}{BC} = \frac{\sin \beta}{\sin \alpha}\).
• From \(\triangle ADC\): \(\frac{AD}{DC} = \frac{\sin \beta}{\sin \alpha}\).

Therefore, a direct consequence of AC bisecting \(\angle A\) and \(\angle C\) is that \(\frac{AB}{BC} = \frac{AD}{DC}\).

Evaluating the Question

The question asks whether \(\frac{AB}{DC} = \frac{AD}{BC}\).

This equality can be rewritten as \(AB \cdot BC = AD \cdot DC\).

We established that \(\frac{AB}{BC} = \frac{AD}{DC}\). Let this common ratio be \(k\). This means \(AB = k \cdot BC\) and \(AD = k \cdot DC\). Substituting these into the equality \(AB \cdot BC = AD \cdot DC\):

Is \((k \cdot BC) \cdot BC = (k \cdot DC) \cdot DC\) ?

This simplifies to \(k \cdot BC^2 = k \cdot DC^2\). Assuming \(k \neq 0\), this equality holds only if \(BC^2 = DC^2\), meaning \(BC = DC\).

Conclusion on Sufficiency

The condition \(BC = DC\) is not guaranteed solely by the fact that AC bisects \(\angle A\) and \(\angle C\). Therefore, the statement \(\frac{AB}{DC} = \frac{AD}{BC}\) is not always true.

Since we can determine that the statement \(\frac{AB}{DC} = \frac{AD}{BC}\) is not necessarily true based only on the given information (AC bisects \(\angle A\) and \(\angle C\)), the question can be answered without needing to refer to Statement I or Statement II.