Triangle ABC geo Triangle ABC is the geometric figure used to demonstrate the triangle inequality theorem. chunk-a5e3dacee89618f913c4948b6ffe64ec unknown_source 1752209912 Triangle Inequality category 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. chunk-a5e3dacee89618f913c4948b6ffe64ec unknown_source 1752209912 Euclid's Fifth Postulate category Euclid's Fifth Postulate is a fundamental principle in geometry, used here to compare angles and sides in the proof. chunk-a5e3dacee89618f913c4948b6ffe64ec unknown_source 1752209912 Proposition 19 category Proposition 19 from Euclid's Elements states that in any triangle, the greater angle is subtended by the greater side. chunk-a5e3dacee89618f913c4948b6ffe64ec unknown_source 1752209912 三角形ABC geo 三角形ABC is the specific triangle used to demonstrate the geometric proof of the triangle inequality theorem. chunk-a5e3dacee89618f913c4948b6ffe64ec unknown_source 1752209912 三角不等式 category 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. chunk-a5e3dacee89618f913c4948b6ffe64ec unknown_source 1752209912 欧几里得第五公理 category 欧几里得第五公理(Euclid's Fifth Postulate) is a classical geometric principle used in this proof to compare angles. chunk-a5e3dacee89618f913c4948b6ffe64ec unknown_source 1752209912 几何原本 category 几何原本(Elements of Geometry) is Euclid's foundational mathematical work containing Proposition 19, referenced in this proof. chunk-a5e3dacee89618f913c4948b6ffe64ec unknown_source 1752209912 命题19 category 命题19 (Proposition 19) states that in any triangle, the greater angle is subtended by the greater side, used in this proof. chunk-a5e3dacee89618f913c4948b6ffe64ec unknown_source 1752209912 点D geo 点D (Point D) is an auxiliary point constructed in the proof by extending side AB to create an isosceles triangle. chunk-a5e3dacee89618f913c4948b6ffe64ec unknown_source 1752209912 等腰三角形BCD geo 等腰三角形BCD (Isosceles Triangle BCD) is formed in the proof construction, showing equal angles at its base. chunk-a5e3dacee89618f913c4948b6ffe64ec unknown_source 1752209912 9.0 Triangle ABC is used to demonstrate the triangle inequality theorem, showing the relationship between its sides. geometric proof,inequality chunk-a5e3dacee89618f913c4948b6ffe64ec unknown_source 1752209912 7.0 Proposition 19 is applied to prove the triangle inequality by comparing angles and corresponding sides. geometric logic,proof technique chunk-a5e3dacee89618f913c4948b6ffe64ec unknown_source 1752209912 8.0 Euclid's Fifth Postulate is used alongside Proposition 19 to establish the relationship between angles and sides in the proof. angle-side relationship,geometric principles chunk-a5e3dacee89618f913c4948b6ffe64ec unknown_source 1752209912 9.0 The proof uses triangle ABC to demonstrate the triangle inequality theorem through geometric construction. geometric proof,inequality demonstration chunk-a5e3dacee89618f913c4948b6ffe64ec unknown_source 1752209912 7.0 Point D is constructed from triangle ABC by extending side AB to create additional geometric relationships. auxiliary point,geometric construction chunk-a5e3dacee89618f913c4948b6ffe64ec unknown_source 1752209912 7.0 The triangle inequality proof references Euclid's Elements (几何原本) as the source of foundational geometric propositions. historical reference,mathematical foundation chunk-a5e3dacee89618f913c4948b6ffe64ec unknown_source 1752209912 8.0 Euclid's Fifth Postulate is used to establish angle comparisons that lead to the application of Proposition 19 in the proof. geometric principles,logical progression chunk-a5e3dacee89618f913c4948b6ffe64ec unknown_source 1752209912 8.0 The isosceles triangle BCD's angle properties enable the application of Proposition 19 regarding angle-side relationships. angle properties,proof technique chunk-a5e3dacee89618f913c4948b6ffe64ec unknown_source 1752209912