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<?xml version='1.0' encoding='utf-8'?>
<graphml xmlns="http://graphml.graphdrawing.org/xmlns" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://graphml.graphdrawing.org/xmlns http://graphml.graphdrawing.org/xmlns/1.0/graphml.xsd">
<key id="d11" for="edge" attr.name="created_at" attr.type="long" />
<key id="d10" for="edge" attr.name="file_path" attr.type="string" />
<key id="d9" for="edge" attr.name="source_id" attr.type="string" />
<key id="d8" for="edge" attr.name="keywords" attr.type="string" />
<key id="d7" for="edge" attr.name="description" attr.type="string" />
<key id="d6" for="edge" attr.name="weight" attr.type="double" />
<key id="d5" for="node" attr.name="created_at" attr.type="long" />
<key id="d4" for="node" attr.name="file_path" attr.type="string" />
<key id="d3" for="node" attr.name="source_id" attr.type="string" />
<key id="d2" for="node" attr.name="description" attr.type="string" />
<key id="d1" for="node" attr.name="entity_type" attr.type="string" />
<key id="d0" for="node" attr.name="entity_id" attr.type="string" />
<graph edgedefault="undirected">
<node id="三角形ABC">
<data key="d0">三角形ABC</data>
<data key="d1">category</data>
<data key="d2">A geometric figure used in the proof of the triangle inequality and angle relationships.</data>
<data key="d3">chunk-4ed981658aa2a32c616f4cf86c4efbaa</data>
<data key="d4">JiHe.docx</data>
<data key="d5">1753235747</data>
</node>
<node id="三角不等式">
<data key="d0">三角不等式</data>
<data key="d1">category</data>
<data key="d2">A mathematical inequality stating that the sum of any two sides of a triangle is greater than the third side.</data>
<data key="d3">chunk-4ed981658aa2a32c616f4cf86c4efbaa</data>
<data key="d4">JiHe.docx</data>
<data key="d5">1753235747</data>
</node>
<node id="欧几里得第五公理">
<data key="d0">欧几里得第五公理</data>
<data key="d1">category</data>
<data key="d2">A fundamental postulate in Euclidean geometry, also known as the parallel postulate.</data>
<data key="d3">chunk-4ed981658aa2a32c616f4cf86c4efbaa</data>
<data key="d4">JiHe.docx</data>
<data key="d5">1753235747</data>
</node>
<node id="几何原本">
<data key="d0">几何原本</data>
<data key="d1">category</data>
<data key="d2">A foundational mathematical text by Euclid, containing definitions, postulates, and propositions.</data>
<data key="d3">chunk-4ed981658aa2a32c616f4cf86c4efbaa</data>
<data key="d4">JiHe.docx</data>
<data key="d5">1753235747</data>
</node>
<node id="点D">
<data key="d0">点D</data>
<data key="d1">category</data>
<data key="d2">A constructed point in the proof, used to demonstrate the triangle inequality.</data>
<data key="d3">chunk-4ed981658aa2a32c616f4cf86c4efbaa</data>
<data key="d4">JiHe.docx</data>
<data key="d5">1753235747</data>
</node>
<node id="点P">
<data key="d0">点P</data>
<data key="d1">category</data>
<data key="d2">An arbitrary internal point in triangle ABC, used to prove angle relationships.</data>
<data key="d3">chunk-4ed981658aa2a32c616f4cf86c4efbaa</data>
<data key="d4">JiHe.docx</data>
<data key="d5">1753235747</data>
</node>
<node id="直线AB">
<data key="d0">直线AB</data>
<data key="d1">category</data>
<data key="d2">Extended line segment in the construction for proving triangle inequality.</data>
<data key="d3">chunk-4ed981658aa2a32c616f4cf86c4efbaa</data>
<data key="d4">JiHe.docx</data>
<data key="d5">1753235747</data>
</node>
<node id="线段DC">
<data key="d0">线段DC</data>
<data key="d1">category</data>
<data key="d2">Constructed line segment forming triangle BCD in the proof.</data>
<data key="d3">chunk-4ed981658aa2a32c616f4cf86c4efbaa</data>
<data key="d4">JiHe.docx</data>
<data key="d5">1753235747</data>
</node>
<node id="角∠BDC">
<data key="d0">角∠BDC</data>
<data key="d1">category</data>
<data key="d2">Angle in isosceles triangle BCD equal to ∠BCD.</data>
<data key="d3">chunk-4ed981658aa2a32c616f4cf86c4efbaa</data>
<data key="d4">JiHe.docx</data>
<data key="d5">1753235752</data>
</node>
<node id="角∠ACD">
<data key="d0">角∠ACD</data>
<data key="d1">category</data>
<data key="d2">Angle compared to ∠ADC in the inequality proof.</data>
<data key="d3">chunk-4ed981658aa2a32c616f4cf86c4efbaa</data>
<data key="d4">JiHe.docx</data>
<data key="d5">1753235753</data>
</node>
<node id="边AD">
<data key="d0">边AD</data>
<data key="d1">category</data>
<data key="d2">Constructed side demonstrating the triangle inequality |AD|&gt;|AC|.</data>
<data key="d3">chunk-4ed981658aa2a32c616f4cf86c4efbaa</data>
<data key="d4">JiHe.docx</data>
<data key="d5">1753235753</data>
</node>
<node id="边AC">
<data key="d0">边AC</data>
<data key="d1">category</data>
<data key="d2">Triangle side compared in the inequality proof.</data>
<data key="d3">chunk-4ed981658aa2a32c616f4cf86c4efbaa</data>
<data key="d4">JiHe.docx</data>
<data key="d5">1753235753</data>
</node>
<node id="BP">
<data key="d0">BP</data>
<data key="d1">organization</data>
<data key="d2">BP is a line segment in the geometric problem, intersecting AC at point D.</data>
<data key="d3">chunk-f24190139d17b1c4d499b9516ffb8a4f</data>
<data key="d4">JiHe.docx</data>
<data key="d5">1753235753</data>
</node>
<node id="AC">
<data key="d0">AC</data>
<data key="d1">organization</data>
<data key="d2">AC is a line segment in the geometric problem, intersected by BP at point D.</data>
<data key="d3">chunk-f24190139d17b1c4d499b9516ffb8a4f</data>
<data key="d4">JiHe.docx</data>
<data key="d5">1753235753</data>
</node>
<node id="D">
<data key="d0">D</data>
<data key="d1">person</data>
<data key="d2">Point D is the intersection point of BP and AC in the geometric problem.</data>
<data key="d3">chunk-f24190139d17b1c4d499b9516ffb8a4f</data>
<data key="d4">JiHe.docx</data>
<data key="d5">1753235756</data>
</node>
<node id="∠BPC">
<data key="d0">∠BPC</data>
<data key="d1">category</data>
<data key="d2">∠BPC is an angle in the geometric problem, which is an exterior angle of triangle PCD.</data>
<data key="d3">chunk-f24190139d17b1c4d499b9516ffb8a4f</data>
<data key="d4">JiHe.docx</data>
<data key="d5">1753235756</data>
</node>
<node id="∠PCD">
<data key="d0">∠PCD</data>
<data key="d1">category</data>
<data key="d2">∠PCD is an angle in the geometric problem, part of triangle PCD.</data>
<data key="d3">chunk-f24190139d17b1c4d499b9516ffb8a4f</data>
<data key="d4">JiHe.docx</data>
<data key="d5">1753235759</data>
</node>
<node id="∠PDC">
<data key="d0">∠PDC</data>
<data key="d1">category</data>
<data key="d2">∠PDC is an angle in the geometric problem, which is an exterior angle of triangle BAD.</data>
<data key="d3">chunk-f24190139d17b1c4d499b9516ffb8a4f</data>
<data key="d4">JiHe.docx</data>
<data key="d5">1753235759</data>
</node>
<node id="∠DBA">
<data key="d0">∠DBA</data>
<data key="d1">category</data>
<data key="d2">∠DBA is an angle in the geometric problem, part of triangle BAD.</data>
<data key="d3">chunk-f24190139d17b1c4d499b9516ffb8a4f</data>
<data key="d4">JiHe.docx</data>
<data key="d5">1753235760</data>
</node>
<node id="∠A">
<data key="d0">∠A</data>
<data key="d1">category</data>
<data key="d2">∠A is an angle in the geometric problem, part of triangle BAD.</data>
<data key="d3">chunk-f24190139d17b1c4d499b9516ffb8a4f</data>
<data key="d4">JiHe.docx</data>
<data key="d5">1753235760</data>
</node>
<node id="△PCD">
<data key="d0">△PCD</data>
<data key="d1">category</data>
<data key="d2">Triangle PCD is a geometric figure in the proof where ∠BPC is an exterior angle.</data>
<data key="d3">chunk-f24190139d17b1c4d499b9516ffb8a4f</data>
<data key="d4">JiHe.docx</data>
<data key="d5">1753235760</data>
</node>
<node id="△BAD">
<data key="d0">△BAD</data>
<data key="d1">category</data>
<data key="d2">Triangle BAD is a geometric figure in the proof where ∠PDC is an exterior angle.</data>
<data key="d3">chunk-f24190139d17b1c4d499b9516ffb8a4f</data>
<data key="d4">JiHe.docx</data>
<data key="d5">1753235760</data>
</node>
<node id="三角形BCD">
<data key="d0">三角形BCD</data>
<data key="d3">chunk-4ed981658aa2a32c616f4cf86c4efbaa</data>
<data key="d2">Segment DC forms one side of the constructed isosceles triangle BCD.</data>
<data key="d1">UNKNOWN</data>
<data key="d4">JiHe.docx</data>
<data key="d5">1753235766</data>
</node>
<node id="角∠BCD">
<data key="d0">角∠BCD</data>
<data key="d3">chunk-4ed981658aa2a32c616f4cf86c4efbaa</data>
<data key="d2">These angles are equal in measure due to the isosceles property of triangle BCD.</data>
<data key="d1">UNKNOWN</data>
<data key="d4">JiHe.docx</data>
<data key="d5">1753235766</data>
</node>
<node id="角∠ADC">
<data key="d0">角∠ADC</data>
<data key="d3">chunk-4ed981658aa2a32c616f4cf86c4efbaa</data>
<data key="d2">Euclid's fifth postulate establishes that ∠ACD &gt; ∠ADC in this configuration.</data>
<data key="d1">UNKNOWN</data>
<data key="d4">JiHe.docx</data>
<data key="d5">1753235771</data>
</node>
<edge source="三角形ABC" target="三角不等式">
<data key="d6">8.0</data>
<data key="d7">The triangle ABC is used to demonstrate the triangle inequality, showing |AB|+|BC|&gt;|AC|.</data>
<data key="d8">geometric proof,inequality</data>
<data key="d9">chunk-4ed981658aa2a32c616f4cf86c4efbaa</data>
<data key="d10">JiHe.docx</data>
<data key="d11">1753235760</data>
</edge>
<edge source="三角形ABC" target="点D">
<data key="d6">6.0</data>
<data key="d7">Point D is constructed within the proof to extend AB and create an isosceles triangle, aiding in the demonstration of the triangle inequality.</data>
<data key="d8">construction,geometric manipulation</data>
<data key="d9">chunk-4ed981658aa2a32c616f4cf86c4efbaa</data>
<data key="d10">JiHe.docx</data>
<data key="d11">1753235766</data>
</edge>
<edge source="三角形ABC" target="点P">
<data key="d6">7.0</data>
<data key="d7">Point P is an arbitrary internal point used to prove that ∠BPC &gt; ∠A, demonstrating angle relationships within the triangle.</data>
<data key="d8">angle proof,internal point</data>
<data key="d9">chunk-4ed981658aa2a32c616f4cf86c4efbaa</data>
<data key="d10">JiHe.docx</data>
<data key="d11">1753235766</data>
</edge>
<edge source="三角不等式" target="欧几里得第五公理">
<data key="d6">7.0</data>
<data key="d7">Euclid's fifth postulate is applied in the proof to establish angle relationships supporting the triangle inequality.</data>
<data key="d8">angle comparison,mathematical foundation</data>
<data key="d9">chunk-4ed981658aa2a32c616f4cf86c4efbaa</data>
<data key="d10">JiHe.docx</data>
<data key="d11">1753235760</data>
</edge>
<edge source="三角不等式" target="几何原本">
<data key="d6">9.0</data>
<data key="d7">The proof references Proposition 19 from Euclid's Elements to justify the relationship between angles and sides.</data>
<data key="d8">geometric principles,historical reference</data>
<data key="d9">chunk-4ed981658aa2a32c616f4cf86c4efbaa</data>
<data key="d10">JiHe.docx</data>
<data key="d11">1753235766</data>
</edge>
<edge source="欧几里得第五公理" target="几何原本">
<data key="d6">8.0</data>
<data key="d7">The fifth postulate is one of the foundational elements in Euclid's Elements.</data>
<data key="d8">axiomatic system,mathematical foundation</data>
<data key="d9">chunk-4ed981658aa2a32c616f4cf86c4efbaa</data>
<data key="d10">JiHe.docx</data>
<data key="d11">1753235766</data>
</edge>
<edge source="点D" target="直线AB">
<data key="d6">7.0</data>
<data key="d7">Point D is constructed by extending line AB to create the proof's geometric configuration.</data>
<data key="d8">geometric construction,proof technique</data>
<data key="d9">chunk-4ed981658aa2a32c616f4cf86c4efbaa</data>
<data key="d10">JiHe.docx</data>
<data key="d11">1753235766</data>
</edge>
<edge source="线段DC" target="三角形BCD">
<data key="d6">8.0</data>
<data key="d7">Segment DC forms one side of the constructed isosceles triangle BCD.</data>
<data key="d8">auxiliary construction,geometric formation</data>
<data key="d9">chunk-4ed981658aa2a32c616f4cf86c4efbaa</data>
<data key="d10">JiHe.docx</data>
<data key="d11">1753235766</data>
</edge>
<edge source="角∠BDC" target="角∠BCD">
<data key="d6">9.0</data>
<data key="d7">These angles are equal in measure due to the isosceles property of triangle BCD.</data>
<data key="d8">angle equality,isosceles properties</data>
<data key="d9">chunk-4ed981658aa2a32c616f4cf86c4efbaa</data>
<data key="d10">JiHe.docx</data>
<data key="d11">1753235766</data>
</edge>
<edge source="角∠ACD" target="角∠ADC">
<data key="d6">8.0</data>
<data key="d7">Euclid's fifth postulate establishes that ∠ACD &gt; ∠ADC in this configuration.</data>
<data key="d8">Euclidean principles,angle comparison</data>
<data key="d9">chunk-4ed981658aa2a32c616f4cf86c4efbaa</data>
<data key="d10">JiHe.docx</data>
<data key="d11">1753235771</data>
</edge>
<edge source="边AD" target="边AC">
<data key="d6">9.0</data>
<data key="d7">The constructed length AD demonstrates the triangle inequality by exceeding AC.</data>
<data key="d8">inequality proof,length comparison</data>
<data key="d9">chunk-4ed981658aa2a32c616f4cf86c4efbaa</data>
<data key="d10">JiHe.docx</data>
<data key="d11">1753235772</data>
</edge>
<edge source="BP" target="AC">
<data key="d6">8.0</data>
<data key="d7">BP intersects AC at point D in the geometric problem.</data>
<data key="d8">geometric relationship,intersection</data>
<data key="d9">chunk-f24190139d17b1c4d499b9516ffb8a4f</data>
<data key="d10">JiHe.docx</data>
<data key="d11">1753235772</data>
</edge>
<edge source="BP" target="D">
<data key="d6">8.0</data>
<data key="d7">BP passes through point D, which is the intersection with AC.</data>
<data key="d8">line-point relationship</data>
<data key="d9">chunk-f24190139d17b1c4d499b9516ffb8a4f</data>
<data key="d10">JiHe.docx</data>
<data key="d11">1753235772</data>
</edge>
<edge source="AC" target="D">
<data key="d6">8.0</data>
<data key="d7">AC passes through point D, which is the intersection with BP.</data>
<data key="d8">line-point relationship</data>
<data key="d9">chunk-f24190139d17b1c4d499b9516ffb8a4f</data>
<data key="d10">JiHe.docx</data>
<data key="d11">1753235772</data>
</edge>
<edge source="∠BPC" target="∠PCD">
<data key="d6">9.0</data>
<data key="d7">∠BPC is an exterior angle of triangle PCD, which includes ∠PCD.</data>
<data key="d8">angle relationship,exterior angle</data>
<data key="d9">chunk-f24190139d17b1c4d499b9516ffb8a4f</data>
<data key="d10">JiHe.docx</data>
<data key="d11">1753235772</data>
</edge>
<edge source="∠BPC" target="∠A">
<data key="d6">7.0</data>
<data key="d7">∠BPC is greater than ∠A in the geometric problem.</data>
<data key="d8">angle comparison,inequality</data>
<data key="d9">chunk-f24190139d17b1c4d499b9516ffb8a4f</data>
<data key="d10">JiHe.docx</data>
<data key="d11">1753235772</data>
</edge>
<edge source="∠BPC" target="△PCD">
<data key="d6">9.0</data>
<data key="d7">∠BPC is an exterior angle of triangle PCD in the geometric proof.</data>
<data key="d8">angle-triangle relationship</data>
<data key="d9">chunk-f24190139d17b1c4d499b9516ffb8a4f</data>
<data key="d10">JiHe.docx</data>
<data key="d11">1753235777</data>
</edge>
<edge source="∠PDC" target="∠DBA">
<data key="d6">9.0</data>
<data key="d7">∠PDC is an exterior angle of triangle BAD, which includes ∠DBA.</data>
<data key="d8">angle relationship,exterior angle</data>
<data key="d9">chunk-f24190139d17b1c4d499b9516ffb8a4f</data>
<data key="d10">JiHe.docx</data>
<data key="d11">1753235772</data>
</edge>
<edge source="∠PDC" target="△BAD">
<data key="d6">9.0</data>
<data key="d7">∠PDC is an exterior angle of triangle BAD in the geometric proof.</data>
<data key="d8">angle-triangle relationship</data>
<data key="d9">chunk-f24190139d17b1c4d499b9516ffb8a4f</data>
<data key="d10">JiHe.docx</data>
<data key="d11">1753235778</data>
</edge>
<edge source="△PCD" target="△BAD">
<data key="d6">7.0</data>
<data key="d7">Both triangles share point D and are connected through the intersecting lines BP and AC.</data>
<data key="d8">geometric connection,shared point</data>
<data key="d9">chunk-f24190139d17b1c4d499b9516ffb8a4f</data>
<data key="d10">JiHe.docx</data>
<data key="d11">1753235778</data>
</edge>
</graph>
</graphml>
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