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<data key="d0">Triangle ABC</data>
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<data key="d2">Triangle ABC is the geometric figure used to demonstrate the triangle inequality theorem.</data>
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<data key="d0">Triangle Inequality</data>
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<data key="d2">The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.</data>
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<data key="d0">Euclid's Fifth Postulate</data>
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<data key="d2">Euclid's Fifth Postulate is a fundamental principle in geometry, used here to compare angles and sides in the proof.</data>
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<data key="d0">三角形ABC</data>
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<data key="d0">三角不等式</data>
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<data key="d2">三角不等式(Triangle Inequality) is a fundamental theorem in geometry stating that the sum of any two sides of a triangle must be greater than the third side.</data>
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<data key="d0">欧几里得第五公理</data>
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<data key="d1">category</data>
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<data key="d2">欧几里得第五公理(Euclid's Fifth Postulate) is a classical geometric principle used in this proof to compare angles.</data>
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<data key="d0">几何原本</data>
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<node id="命题19">
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<data key="d0">命题19</data>
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<data key="d1">category</data>
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<data key="d2">命题19 (Proposition 19) states that in any triangle, the greater angle is subtended by the greater side, used in this proof.</data>
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<node id="点D">
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<data key="d0">点D</data>
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<data key="d1">geo</data>
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<data key="d2">点D (Point D) is an auxiliary point constructed in the proof by extending side AB to create an isosceles triangle.</data>
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<data key="d1">geo</data>
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<data key="d2">等腰三角形BCD (Isosceles Triangle BCD) is formed in the proof construction, showing equal angles at its base.</data>
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<data key="d6">9.0</data>
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<edge source="Triangle Inequality" target="Proposition 19">
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<data key="d6">7.0</data>
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<data key="d6">8.0</data>
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<data key="d7">Euclid's Fifth Postulate is used alongside Proposition 19 to establish the relationship between angles and sides in the proof.</data>
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<data key="d7">Point D is constructed from triangle ABC by extending side AB to create additional geometric relationships.</data>
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<data key="d7">The triangle inequality proof references Euclid's Elements (几何原本) as the source of foundational geometric propositions.</data>
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<data key="d6">8.0</data>
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<data key="d7">Euclid's Fifth Postulate is used to establish angle comparisons that lead to the application of Proposition 19 in the proof.</data>
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<data key="d7">The isosceles triangle BCD's angle properties enable the application of Proposition 19 regarding angle-side relationships.</data>
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