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- https://math.libretexts.org/Bookshelves/Geometry/Modern_Geometry_(Bishop)/00%3A_Front_Matter/02%3A_InfoPageThe LibreTexts libraries are Powered by MindTouch ® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the Californ...The LibreTexts libraries are Powered by MindTouch ® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot.
- https://math.libretexts.org/Bookshelves/Geometry/Modern_Geometry_(Bishop)/zz%3A_Back_Matter/10%3A_Index
- https://math.libretexts.org/Bookshelves/Geometry/Modern_Geometry_(Bishop)/01%3A_Euclids_Elements_Props_1-28_of_Book_1/1.02%3A_Book_1_DefinitionsIf two triangles have the two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will ...If two triangles have the two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend. [That is, Side-Angle-Side (SAS) that we will assume as our triangle congruence axiom.
- https://math.libretexts.org/Bookshelves/Geometry/Modern_Geometry_(Bishop)The standard approach is to develop more results of advanced Euclidean geometry first and to eventually back up and go into hyperbolic geometry. In order to get where we need to get, we will not forma...The standard approach is to develop more results of advanced Euclidean geometry first and to eventually back up and go into hyperbolic geometry. In order to get where we need to get, we will not formally develop some of the advanced Euclidean results that are logically needed because we would never get to the "modern" part of the course’s name. We will come back and develop, or at least introduce conceptually, some of these when we need them late in the course.
- https://math.libretexts.org/Bookshelves/Geometry/Modern_Geometry_(Bishop)/04%3A_Elementary_Euclidean_Geometry/4.04%3A_Problem_Set_4Suppose the accompanying figure is the cross-section of the earth determined by the plane of O,S and B; respectively, the center of the earth, the center of the sun...Suppose the accompanying figure is the cross-section of the earth determined by the plane of O,S and B; respectively, the center of the earth, the center of the sun (too far away on the right to be visible), and a point a known distance d directly north of the point A at which the sun is directly overhead (for simplicity, on the equator, at noon on an equinox).
- https://math.libretexts.org/Bookshelves/Geometry/Modern_Geometry_(Bishop)/zz%3A_Back_Matter
- https://math.libretexts.org/Bookshelves/Geometry/Modern_Geometry_(Bishop)/01%3A_Euclids_Elements_Props_1-28_of_Book_1/1.01%3A_Introduction_to_Euclids_ElementsBut FC was also proved equal to GB; therefore the two sides BF,FC are equal to the two sides CG, GB respectively; and the angle BFC is equal to the angle CGB, while the base \(\...But FC was also proved equal to GB; therefore the two sides BF,FC are equal to the two sides CG, GB respectively; and the angle BFC is equal to the angle CGB, while the base BC is common to them; therefore the triangle BFC is also equal to the triangle CGB, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend; therefore the angle FBC is equal to the angle GCB, and the angle BCF to the angle CB…
- https://math.libretexts.org/Bookshelves/Geometry/Modern_Geometry_(Bishop)/00%3A_Front_Matter
- https://math.libretexts.org/Bookshelves/Geometry/Modern_Geometry_(Bishop)/02%3A_More_Elementary_Neutral_Geometry/2.02%3A_Problem_Set_2Prove that the line PT is the only tangent on P to the circle on the same side of line of the OP as point B. [Hint: Let S be a point on the circle on the same side of lin...Prove that the line PT is the only tangent on P to the circle on the same side of line of the OP as point B. [Hint: Let S be a point on the circle on the same side of line OP as T such that line PS is tangent to the circle and prove that T=S, hence exactly one tangent on each side of line OP.]
- https://math.libretexts.org/Bookshelves/Geometry/Modern_Geometry_(Bishop)/03%3A_Introduction_to_Hyperbolic_Geometry/3.02%3A_Problem_Set_3Assuming that the Poincaré disk model really is a model for hyperbolic geometry, recall the construction for the tangents from a point P outside of circle O given in PS 2, #28, and explain the indicat...Assuming that the Poincaré disk model really is a model for hyperbolic geometry, recall the construction for the tangents from a point P outside of circle O given in PS 2, #28, and explain the indicated construction of tangent lines PT and PT′ to the Poincaré circle (O; OA) (i.e., center O of radius OA) in the pictured Poincaré disk model (assuming a "Poincaré compass" and "Poincaré straightedge") and prove that the construction is valid. [Note1: E…
- https://math.libretexts.org/Bookshelves/Geometry/Modern_Geometry_(Bishop)/01%3A_Euclids_Elements_Props_1-28_of_Book_1/1.03%3A_Problem_Set_1Prove that the definite article is correct and argue that on a sphere such a line need not be unique. [Hint: The North Pole and the Equator can be helpful here.] It is "the" under the axioms of neutra...Prove that the definite article is correct and argue that on a sphere such a line need not be unique. [Hint: The North Pole and the Equator can be helpful here.] It is "the" under the axioms of neutral geometry but this is somewhat subtle, a consequence of the Exterior Angle Theorem, Prop.16.]
