Euclidean geometry and proof

इस पर आधारित

21 अनिवार्य पूर्वापेक्षाएँ (बिल्कुल नीचे तक)

आप सीखेंगे

• Use the congruence criteria SAS, ASA and SSS to prove properties of triangles and quadrilaterals.
• Prove, using the parallel postulate, that the angles of a triangle sum to 180°, and prove circle theorems such as the inscribed angle theorem.
• Explain the role of the parallel postulate and how its independence led to non-Euclidean geometries.

परिभाषा

• Parallel postulate (Playfair's form): through a point not on a given line there is exactly one line parallel to the given line.
• Two triangles are congruent if their corresponding sides and corresponding angles are equal.
• An inscribed angle has its vertex on a circle and its sides along chords of the circle.

प्रमेय

• The angles of a triangle sum to 180°; given the other axioms together with Archimedes' axiom, this is equivalent to the parallel postulate (Dehn's semi-Euclidean planes show Archimedes' axiom is needed).
• Inscribed angle theorem: an inscribed angle is half the central angle subtending the same arc.
• The perpendicular bisectors of the sides of a triangle meet in a single point, the circumcentre.

आम भ्रांतियाँ

• Arguing from the appearance of a diagram instead of from stated facts.
• Treating SSA as a congruence criterion.
• Believing the parallel postulate has been proved from Euclid's other axioms; it is independent of them.

आगे ले जाता है

वैकल्पिक खोजें

• First-order logic The parallel postulate is independent of the other axioms: hyperbolic geometry satisfies them while violating it. Tarski showed that elementary Euclidean geometry, formalised in first-order logic, is complete and decidable.
• Differential geometry of curves and surfaces On a sphere of radius r the angles of a triangle sum to more than 180°, and the excess (in radians) equals the area divided by r² (Girard's theorem), a first case of Gauss–Bonnet.
• Projective geometry Adding a point at infinity where each family of parallel lines meets gives the projective plane, in which any two distinct lines meet in exactly one point.

अधिगम पथ

• Conventional K–12 progression
• Geometry and topology

आगे पढ़ने के लिए

• Euclid, Elements
• Robin Hartshorne, Geometry: Euclid and Beyond (2000)
• A. P. Kiselev, adapted by Alexander Givental, Kiselev's Geometry, Book I: Planimetry