三角形ABC category A geometric figure used in the proof of the triangle inequality and angle relationships. chunk-4ed981658aa2a32c616f4cf86c4efbaa JiHe.docx 1753235747 三角不等式 category A mathematical inequality stating that the sum of any two sides of a triangle is greater than the third side. chunk-4ed981658aa2a32c616f4cf86c4efbaa JiHe.docx 1753235747 欧几里得第五公理 category A fundamental postulate in Euclidean geometry, also known as the parallel postulate. chunk-4ed981658aa2a32c616f4cf86c4efbaa JiHe.docx 1753235747 几何原本 category A foundational mathematical text by Euclid, containing definitions, postulates, and propositions. chunk-4ed981658aa2a32c616f4cf86c4efbaa JiHe.docx 1753235747 点D category A constructed point in the proof, used to demonstrate the triangle inequality. chunk-4ed981658aa2a32c616f4cf86c4efbaa JiHe.docx 1753235747 点P category An arbitrary internal point in triangle ABC, used to prove angle relationships. chunk-4ed981658aa2a32c616f4cf86c4efbaa JiHe.docx 1753235747 直线AB category Extended line segment in the construction for proving triangle inequality. chunk-4ed981658aa2a32c616f4cf86c4efbaa JiHe.docx 1753235747 线段DC category Constructed line segment forming triangle BCD in the proof. chunk-4ed981658aa2a32c616f4cf86c4efbaa JiHe.docx 1753235747 角∠BDC category Angle in isosceles triangle BCD equal to ∠BCD. chunk-4ed981658aa2a32c616f4cf86c4efbaa JiHe.docx 1753235752 角∠ACD category Angle compared to ∠ADC in the inequality proof. chunk-4ed981658aa2a32c616f4cf86c4efbaa JiHe.docx 1753235753 边AD category Constructed side demonstrating the triangle inequality |AD|>|AC|. chunk-4ed981658aa2a32c616f4cf86c4efbaa JiHe.docx 1753235753 边AC category Triangle side compared in the inequality proof. chunk-4ed981658aa2a32c616f4cf86c4efbaa JiHe.docx 1753235753 BP organization BP is a line segment in the geometric problem, intersecting AC at point D. chunk-f24190139d17b1c4d499b9516ffb8a4f JiHe.docx 1753235753 AC organization AC is a line segment in the geometric problem, intersected by BP at point D. chunk-f24190139d17b1c4d499b9516ffb8a4f JiHe.docx 1753235753 D person Point D is the intersection point of BP and AC in the geometric problem. chunk-f24190139d17b1c4d499b9516ffb8a4f JiHe.docx 1753235756 ∠BPC category ∠BPC is an angle in the geometric problem, which is an exterior angle of triangle PCD. chunk-f24190139d17b1c4d499b9516ffb8a4f JiHe.docx 1753235756 ∠PCD category ∠PCD is an angle in the geometric problem, part of triangle PCD. chunk-f24190139d17b1c4d499b9516ffb8a4f JiHe.docx 1753235759 ∠PDC category ∠PDC is an angle in the geometric problem, which is an exterior angle of triangle BAD. chunk-f24190139d17b1c4d499b9516ffb8a4f JiHe.docx 1753235759 ∠DBA category ∠DBA is an angle in the geometric problem, part of triangle BAD. chunk-f24190139d17b1c4d499b9516ffb8a4f JiHe.docx 1753235760 ∠A category ∠A is an angle in the geometric problem, part of triangle BAD. chunk-f24190139d17b1c4d499b9516ffb8a4f JiHe.docx 1753235760 △PCD category Triangle PCD is a geometric figure in the proof where ∠BPC is an exterior angle. chunk-f24190139d17b1c4d499b9516ffb8a4f JiHe.docx 1753235760 △BAD category Triangle BAD is a geometric figure in the proof where ∠PDC is an exterior angle. chunk-f24190139d17b1c4d499b9516ffb8a4f JiHe.docx 1753235760 三角形BCD chunk-4ed981658aa2a32c616f4cf86c4efbaa Segment DC forms one side of the constructed isosceles triangle BCD. UNKNOWN JiHe.docx 1753235766 角∠BCD chunk-4ed981658aa2a32c616f4cf86c4efbaa These angles are equal in measure due to the isosceles property of triangle BCD. UNKNOWN JiHe.docx 1753235766 角∠ADC chunk-4ed981658aa2a32c616f4cf86c4efbaa Euclid's fifth postulate establishes that ∠ACD > ∠ADC in this configuration. UNKNOWN JiHe.docx 1753235771 8.0 The triangle ABC is used to demonstrate the triangle inequality, showing |AB|+|BC|>|AC|. geometric proof,inequality chunk-4ed981658aa2a32c616f4cf86c4efbaa JiHe.docx 1753235760 6.0 Point D is constructed within the proof to extend AB and create an isosceles triangle, aiding in the demonstration of the triangle inequality. construction,geometric manipulation chunk-4ed981658aa2a32c616f4cf86c4efbaa JiHe.docx 1753235766 7.0 Point P is an arbitrary internal point used to prove that ∠BPC > ∠A, demonstrating angle relationships within the triangle. angle proof,internal point chunk-4ed981658aa2a32c616f4cf86c4efbaa JiHe.docx 1753235766 7.0 Euclid's fifth postulate is applied in the proof to establish angle relationships supporting the triangle inequality. angle comparison,mathematical foundation chunk-4ed981658aa2a32c616f4cf86c4efbaa JiHe.docx 1753235760 9.0 The proof references Proposition 19 from Euclid's Elements to justify the relationship between angles and sides. geometric principles,historical reference chunk-4ed981658aa2a32c616f4cf86c4efbaa JiHe.docx 1753235766 8.0 The fifth postulate is one of the foundational elements in Euclid's Elements. axiomatic system,mathematical foundation chunk-4ed981658aa2a32c616f4cf86c4efbaa JiHe.docx 1753235766 7.0 Point D is constructed by extending line AB to create the proof's geometric configuration. geometric construction,proof technique chunk-4ed981658aa2a32c616f4cf86c4efbaa JiHe.docx 1753235766 8.0 Segment DC forms one side of the constructed isosceles triangle BCD. auxiliary construction,geometric formation chunk-4ed981658aa2a32c616f4cf86c4efbaa JiHe.docx 1753235766 9.0 These angles are equal in measure due to the isosceles property of triangle BCD. angle equality,isosceles properties chunk-4ed981658aa2a32c616f4cf86c4efbaa JiHe.docx 1753235766 8.0 Euclid's fifth postulate establishes that ∠ACD > ∠ADC in this configuration. Euclidean principles,angle comparison chunk-4ed981658aa2a32c616f4cf86c4efbaa JiHe.docx 1753235771 9.0 The constructed length AD demonstrates the triangle inequality by exceeding AC. inequality proof,length comparison chunk-4ed981658aa2a32c616f4cf86c4efbaa JiHe.docx 1753235772 8.0 BP intersects AC at point D in the geometric problem. geometric relationship,intersection chunk-f24190139d17b1c4d499b9516ffb8a4f JiHe.docx 1753235772 8.0 BP passes through point D, which is the intersection with AC. line-point relationship chunk-f24190139d17b1c4d499b9516ffb8a4f JiHe.docx 1753235772 8.0 AC passes through point D, which is the intersection with BP. line-point relationship chunk-f24190139d17b1c4d499b9516ffb8a4f JiHe.docx 1753235772 9.0 ∠BPC is an exterior angle of triangle PCD, which includes ∠PCD. angle relationship,exterior angle chunk-f24190139d17b1c4d499b9516ffb8a4f JiHe.docx 1753235772 7.0 ∠BPC is greater than ∠A in the geometric problem. angle comparison,inequality chunk-f24190139d17b1c4d499b9516ffb8a4f JiHe.docx 1753235772 9.0 ∠BPC is an exterior angle of triangle PCD in the geometric proof. angle-triangle relationship chunk-f24190139d17b1c4d499b9516ffb8a4f JiHe.docx 1753235777 9.0 ∠PDC is an exterior angle of triangle BAD, which includes ∠DBA. angle relationship,exterior angle chunk-f24190139d17b1c4d499b9516ffb8a4f JiHe.docx 1753235772 9.0 ∠PDC is an exterior angle of triangle BAD in the geometric proof. angle-triangle relationship chunk-f24190139d17b1c4d499b9516ffb8a4f JiHe.docx 1753235778 7.0 Both triangles share point D and are connected through the intersecting lines BP and AC. geometric connection,shared point chunk-f24190139d17b1c4d499b9516ffb8a4f JiHe.docx 1753235778