三角形ABC category A geometric figure used to demonstrate the triangle inequality theorem and other geometric properties. chunk-d4c4f366a47f3e13da193e0b600addae unknown_source 1752656859 三角不等式 category A mathematical inequality stating that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. chunk-d4c4f366a47f3e13da193e0b600addae unknown_source 1752656859 欧几里得第五公理 category A foundational axiom in Euclidean geometry, also known as the parallel postulate, used in geometric proofs. chunk-d4c4f366a47f3e13da193e0b600addae unknown_source 1752656859 几何原本 category A classical mathematical text by Euclid, containing foundational principles of geometry, including propositions like 'the larger angle subtends the larger side.' chunk-d4c4f366a47f3e13da193e0b600addae unknown_source 1752656859 点D category A constructed point in the geometric proof used to demonstrate the triangle inequality by forming an isosceles triangle. chunk-d4c4f366a47f3e13da193e0b600addae unknown_source 1752656859 点P category An arbitrary interior point of triangle ABC used to prove that ∠BPC > ∠A through geometric relationships. chunk-d4c4f366a47f3e13da193e0b600addae unknown_source 1752656859 直线AB category A line segment in triangle ABC extended to point D in the proof of the triangle inequality. chunk-d4c4f366a47f3e13da193e0b600addae unknown_source 1752656859 等腰三角形BCD category An isosceles triangle formed in the proof, where |BD|=|BC| and ∠BDC=∠BCD. chunk-d4c4f366a47f3e13da193e0b600addae unknown_source 1752656859 命题19 category A proposition from Euclid's Elements stating that the larger angle subtends the larger side, used in the proof. chunk-d4c4f366a47f3e13da193e0b600addae unknown_source 1752656859 BP organization BP is a line segment in the geometric figure, intersecting AC at point D. chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859 AC organization AC is a line segment in the geometric figure, intersected by BP at point D. chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859 D person D is the point of intersection between BP and AC in the geometric figure. chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859 ∠BPC category ∠BPC is an angle in the geometric figure, which is an exterior angle of triangle PCD. chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859 ∠PCD category ∠PCD is an angle in the geometric figure, part of triangle PCD. chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859 ∠PDC category ∠PDC is an angle in the geometric figure, part of triangle PCD and an exterior angle of triangle BAD. chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859 ∠DBA category ∠DBA is an angle in the geometric figure, part of triangle BAD. chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859 ∠A category ∠A is an angle in the geometric figure, part of triangle BAD. chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859 P person Point P is a vertex in the geometric proof where angle BPC is located. chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859 B person Point B is a vertex in the geometric proof where angle DBA is located. chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859 A person Point A is a vertex in the geometric proof where angle A is located. chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859 C person Point C is a vertex in the geometric proof where angle PCD is located. chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859 △PCD category Triangle PCD is part of the geometric proof where angle BPC is an exterior angle. chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859 △BAD category Triangle BAD is part of the geometric proof where angle PDC is an exterior angle. chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859 9.0 The triangle ABC is used to demonstrate the triangle inequality, showing that the sum of two sides is greater than the third. geometric proof,inequality chunk-d4c4f366a47f3e13da193e0b600addae unknown_source 1752656859 7.0 The proof involving triangle ABC relies on propositions from Euclid's 'Elements,' such as the relationship between angles and sides. classical text,geometric principles chunk-d4c4f366a47f3e13da193e0b600addae unknown_source 1752656859 8.0 Point D is constructed by extending side AB of triangle ABC to demonstrate the triangle inequality. geometric construction,proof technique chunk-d4c4f366a47f3e13da193e0b600addae unknown_source 1752656859 7.0 Point P is an arbitrary interior point of triangle ABC used to prove that ∠BPC > ∠A. geometric proof,interior angle chunk-d4c4f366a47f3e13da193e0b600addae unknown_source 1752656859 8.0 Euclid's Fifth Postulate is a key component of the 'Elements of Geometry,' which forms the basis for many geometric proofs. axiomatic system,mathematical foundation chunk-d4c4f366a47f3e13da193e0b600addae unknown_source 1752656859 8.0 The properties of isosceles triangle BCD are used in conjunction with Euclid's Fifth Postulate to prove the triangle inequality. axiomatic application,proof logic chunk-d4c4f366a47f3e13da193e0b600addae unknown_source 1752656859 9.0 Point D is connected to C to form the isosceles triangle BCD, where |BD|=|BC|. geometric construction,isosceles properties chunk-d4c4f366a47f3e13da193e0b600addae unknown_source 1752656859 9.0 Line AB is extended to create point D, which is essential for constructing the proof of the triangle inequality. line extension,proof construction chunk-d4c4f366a47f3e13da193e0b600addae unknown_source 1752656859 8.0 The proof involving point P uses Proposition 19 from Euclid's Elements to compare angles and sides. angle comparison,geometric proposition chunk-d4c4f366a47f3e13da193e0b600addae unknown_source 1752656859 8.0 BP intersects AC at point D in the geometric figure. geometric intersection,line segments chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859 8.0 BP passes through point D where it intersects AC. intersection,line-point relation chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859 8.0 AC passes through point D where it intersects BP. intersection,line-point relation chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859 9.0 Point D is a vertex of triangle PCD. geometric construction,triangle vertex chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859 9.0 Point D is a vertex of triangle BAD. geometric construction,triangle vertex chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859 9.0 ∠BPC is an exterior angle of triangle PCD, which includes ∠PCD. angle relationship,exterior angle chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859 9.0 ∠BPC is an exterior angle of triangle PCD, which includes ∠PDC. angle relationship,exterior angle chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859 10.0 ∠BPC is greater than ∠A due to the sum of angles ∠PCD and ∠DBA. angle comparison,geometric proof chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859 10.0 Angle BPC is an exterior angle of triangle PCD. exterior angle,triangle property chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859 10.0 Angle PCD is an interior angle of triangle PCD. interior angle,triangle property chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859 9.0 ∠PDC is an exterior angle of triangle BAD, which includes ∠DBA. angle relationship,exterior angle chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859 9.0 ∠PDC is an exterior angle of triangle BAD, which includes ∠A. angle relationship,exterior angle chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859 10.0 Angle PDC is an interior angle of triangle PCD. interior angle,triangle property chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859 10.0 Angle PDC is an exterior angle of triangle BAD. exterior angle,triangle property chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859 10.0 Angle DBA is an interior angle of triangle BAD. interior angle,triangle property chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859 10.0 Angle A is an interior angle of triangle BAD. interior angle,triangle property chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859 9.0 Point P is a vertex of triangle PCD. geometric construction,triangle vertex chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859 9.0 Point B is a vertex of triangle BAD. geometric construction,triangle vertex chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859 9.0 Point A is a vertex of triangle BAD. geometric construction,triangle vertex chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859 9.0 Point C is a vertex of triangle PCD. geometric construction,triangle vertex chunk-618195be0ad6c05ea189e352a4c2e7bb unknown_source 1752656859