Prove that the angle bisectors of a triangle are concurrent.
Proof:
Let \(ABC\) be the triangle.
Let \(AD\), \(BE\) and \(CF\) be the angle bisectors of the triangle \(ABC\) respectively intersecting at \(O\).
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By the Angle Bisector Theorem, we have:
\(AD\) is the angle bisector of
So, \(\frac{AB}{AC} =\) …… \((1)\)
\(BE\) is the angle bisector of .
So, \(\frac{BC}{AB} =\) …… \((2)\)
\(CF\) is the angle bisector of .
So, \(\frac{AC}{BC} =\) …… \((3)\)
Multiply equations \((1)\), \((2)\) and \((3)\).
\(\frac{AB}{AC} \times \frac{BC}{AB} \times \frac{AC}{BC} = \frac{BD}{DC} \times \frac{CE}{EA} \times \frac{AF}{FB}\)
\(\Rightarrow\) \(\frac{BD}{DC} \times \frac{CE}{EA} \times \frac{AF}{FB} = 1\)
Thus, by , the angle bisectors \(AD\), \(BE\) and \(CF\) are congruent.
Therefore, the angle bisectors of a triangle are concurrent.
Hence, it proved.