In a triangle ABC, D is a point on BC. If AB \cdot DC = AC \cdot BD, \angle BAD = \alpha and \angle CAD = \beta then which one of the following is correct?
- A. \alpha=\beta ✓
- B. \alpha=2\beta
- C. 2\alpha=\beta
- D. 2\alpha=3\beta
Correct Answer: A. \alpha=\beta
Explanation
Rearranging the given equation gives \frac{AB}{AC} = \frac{BD}{DC}. According to the converse of the Angle Bisector Theorem, if a line segment AD divides the opposite side into segments proportional to the adjacent sides, then it bisects the interior angle. Thus, AD is the angle bisector of \angle BAC, which means \alpha = \beta.
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