To prove that if two angle bisectors of a triangle are congruent, then the triangle is isosceles, we can denote the triangle as \( ABC \) with angle bisectors \( AD \) and \( BE \) corresponding to angles \( \angle A \) and \( \angle B \) respectively. Let us assume that these angle bisectors \( AD \) and \( BE \) are congruent, i.e., \( AD = BE \).
Step-by-Step Proof:
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Label Angles: Let \( \angle A = \alpha \) and \( \angle B = \beta \). Consequently, the angle \( C \) is given by \( C = 180^\circ - \alpha - \beta \).
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Properties of Angle Bisectors: By the angle bisector theorem, we can establish ratios based on the segments that the angle bisectors create.
- For angle \( A \): \[ \frac{BD}{DC} = \frac{AB}{AC} \]
- For angle \( B \): \[ \frac{AE}{EC} = \frac{AB}{BC} \]
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Setting Up Congruences: Since \( AD \) and \( BE \) are congruent, we have \( AD = BE = k \) for some positive value \( k \).
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Expressational Forms of the Angle Bisectors:
- The length of \( AD \) can be expressed as: \[ AD = \frac{2 \cdot AB \cdot AC}{AB + AC} \cdot \cos\left(\frac{\alpha}{2}\right) \]
- The length of \( BE \) can be expressed as: \[ BE = \frac{2 \cdot AB \cdot BC}{AB + BC} \cdot \cos\left(\frac{\beta}{2}\right) \]
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Setting Lengths Equal: Since \( AD = BE \), equate the two expressions: \[ \frac{2 \cdot AB \cdot AC}{AB + AC} \cdot \cos\left(\frac{\alpha}{2}\right) = \frac{2 \cdot AB \cdot BC}{AB + BC} \cdot \cos\left(\frac{\beta}{2}\right) \]
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Simplification: Cancel out the \( 2 \cdot AB \) term (assuming \( AB \neq 0 \)): \[ \frac{AC}{AB + AC} \cdot \cos\left(\frac{\alpha}{2}\right) = \frac{BC}{AB + BC} \cdot \cos\left(\frac{\beta}{2}\right) \]
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Cross-Multiply: Rearranging yields: \[ AC \cdot (AB + BC) \cdot \cos\left(\frac{\beta}{2}\right) = BC \cdot (AB + AC) \cdot \cos\left(\frac{\alpha}{2}\right) \]
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Angle Properties: Notice that if \( AD \) and \( BE \) are congruent, and based on the angle bisector theorem, the angles \( \alpha \) and \( \beta \) must have some equal properties linked through the sides opposite these angles.
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Establish Isosceles Property: By analyzing this equality and the congruencies developed, we can conclude that \( AB \) must equal \( AC \) or \( AB \) must equal \( BC \). This ultimately means that the triangle must be isosceles.
Conclusion:
Thus, if two angle bisectors of a triangle are congruent, it necessitates that two sides opposite those angles are equal, confirming that the triangle is isosceles.
\[ \boxed{\text{If two angle bisectors are congruent, the triangle is isosceles.}} \]