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- https://math.libretexts.org/Bookshelves/Geometry/Euclidean_Plane_and_its_Relatives_(Petrunin)/12%3A_Hyperbolic_Lane/12.06%3A_Axiom_IIINote that the first part of Axiom III follows directly from the definition of the h-angle measure defined on page . It remains to show that \(\measuredangle_h\) satisfies the conditions Axiom IIIa, Ax...Note that the first part of Axiom III follows directly from the definition of the h-angle measure defined on page . It remains to show that \(\measuredangle_h\) satisfies the conditions Axiom IIIa, Axiom IIIb, and Axiom IIIc. Since \(\measuredangle_h P' O R'=\measuredangle P' O R'\) and the maps \((Q,P)\mapsto P'\), \((Q,R)\mapsto R'\) are continuous, the claim follows from the corresponding axiom of the Euclidean plane.
- https://math.libretexts.org/Bookshelves/Geometry/Euclidean_Plane_and_its_Relatives_(Petrunin)/20%3A_Area/20.05%3A_Section_5-\(\begin{aligned} s(a,b)&=s(a',b)+s(a-a',b)\ge \\ &\ge s(a',b)= \\ &\ge s(a',b')+s(a',b-b')\ge \\ &\ge s(a',b').\end{aligned}\) \(\begin{aligned} s(\tfrac kl,\tfrac mn)&=k \cdot s(\tfrac 1l,\tfrac mn)...\(\begin{aligned} s(a,b)&=s(a',b)+s(a-a',b)\ge \\ &\ge s(a',b)= \\ &\ge s(a',b')+s(a',b-b')\ge \\ &\ge s(a',b').\end{aligned}\) \(\begin{aligned} s(\tfrac kl,\tfrac mn)&=k \cdot s(\tfrac 1l,\tfrac mn)= \\ &=k\cdot m \cdot s(\tfrac 1l,\tfrac 1n)= \\ &=k\cdot m\cdot \tfrac 1l\cdot s(1, \tfrac 1n)= \\ &=k\cdot m\cdot \tfrac 1l\cdot \tfrac 1n\cdot s(1,1)= \\ &=\tfrac kl\cdot\tfrac mn\end{aligned}\)
- https://math.libretexts.org/Bookshelves/Geometry/Euclidean_Plane_and_its_Relatives_(Petrunin)/01%3A_Preliminaries/1.02%3A_What_is_a_modelDefine a point in the Euclidean plane as a pair of real numbers \((x, y)\) and define the distance between the two points \((x_1, y_1)\) and \((x_2, y_2)\) by the following formula: We gave a numerica...Define a point in the Euclidean plane as a pair of real numbers \((x, y)\) and define the distance between the two points \((x_1, y_1)\) and \((x_2, y_2)\) by the following formula: We gave a numerical model of the Euclidean plane; it builds the Euclidean plane from the real numbers while the latter is assumed to be known. On the other hand, the observations made in the previous section are intuitively obvious — this is the main advantage of the axiomatic approach.
- https://math.libretexts.org/Bookshelves/Geometry/Euclidean_Plane_and_its_Relatives_(Petrunin)/04%3A_Congruent_Triangles/4.01%3A_Side-Angle-Side_ConditionOur next goal is to give conditions that guarantee congruence of two triangles. One of such conditions is given in Axiom IV; it states that if two pairs of sides of two triangles are equal, and the in...Our next goal is to give conditions that guarantee congruence of two triangles. One of such conditions is given in Axiom IV; it states that if two pairs of sides of two triangles are equal, and the included angles are equal up to sign, then the triangles are congruent. This axiom is also called side- angle-side congruence condition, or briefly, SAS congruence condition.
- https://math.libretexts.org/Bookshelves/Geometry/Euclidean_Plane_and_its_Relatives_(Petrunin)/09%3A_Inscribed_angles/9.04%3A_Method_of_additional_circle\(\begin{array} {rcl} {2 \cdot \measuredangle CXY \equiv 2 \cdot \measuredangle CPY} & \ \ \ \ & {2 \cdot \measuredangle BXZ \equiv 2 \cdot \measuredangle BPZ,} \\ {2 \cdot \measuredangle YAZ \equiv 2...\(\begin{array} {rcl} {2 \cdot \measuredangle CXY \equiv 2 \cdot \measuredangle CPY} & \ \ \ \ & {2 \cdot \measuredangle BXZ \equiv 2 \cdot \measuredangle BPZ,} \\ {2 \cdot \measuredangle YAZ \equiv 2 \cdot \measuredangle YPZ} & \ \ \ \ & {2 \cdot \measuredangle CAB \equiv 2 \cdot \measuredangle CPB.} \end{array}\)
- https://math.libretexts.org/Bookshelves/Geometry/Euclidean_Plane_and_its_Relatives_(Petrunin)/16%3A_Spherical_geometry/16.01%3A_Euclidean_spaceRecall that Euclidean space is the set \(\mathbb{R}^3\) of all triples \((x,y,z)\) of real numbers such that the distance between a pair of points \(A=(x_A,y_A,z_A)\) and \(B=(x_B,y_B,z_B)\) is define...Recall that Euclidean space is the set \(\mathbb{R}^3\) of all triples \((x,y,z)\) of real numbers such that the distance between a pair of points \(A=(x_A,y_A,z_A)\) and \(B=(x_B,y_B,z_B)\) is defined by the following formula: Formally, sphere with center \(O\) and radius \(r\) is the set of points in the space that lie on the distance \(r\) from \(O\). In spherical geometry, the role of lines play the great circles; that is, the intersection of the sphere with a plane passing thru \(O\).
- https://math.libretexts.org/Bookshelves/Geometry/Euclidean_Plane_and_its_Relatives_(Petrunin)/10%3A_Inversion/10.05%3A_Perpendicular_circlesSimilarly, we say that the circle \(\Gamma\) is perpendicular to the line \(\ell\) (briefly \(\Gamma \perp \ell\)) if \(\Gamma \cap \ell \ne \emptyset\) and \(\ell\) perpendicular to the tangent lines...Similarly, we say that the circle \(\Gamma\) is perpendicular to the line \(\ell\) (briefly \(\Gamma \perp \ell\)) if \(\Gamma \cap \ell \ne \emptyset\) and \(\ell\) perpendicular to the tangent lines to \(\Gamma\) at one point (and therefore, both points) of intersection.
- https://math.libretexts.org/Bookshelves/Geometry/Euclidean_Plane_and_its_Relatives_(Petrunin)/01%3A_Preliminaries/1.07%3A_AnglesIntuitively, the angle measure tells how much one has to rotate the first half-line counterclockwise, so it gets the position of the second half-line of the angle. The full turn is assumed to be \(2 \...Intuitively, the angle measure tells how much one has to rotate the first half-line counterclockwise, so it gets the position of the second half-line of the angle. The full turn is assumed to be \(2 \cdot \pi\); it corresponds to the angle measure in radians. (For a while you may think that \(\pi\) is a positive real number that measures the size of a half turn in certain units.
- https://math.libretexts.org/Bookshelves/Geometry/Euclidean_Plane_and_its_Relatives_(Petrunin)/16%3A_Spherical_geometry/16.02%3A_Pythagorean_TheoremHere is an analog of the Pythagorean theorems in spherical geometry.
- https://math.libretexts.org/Bookshelves/Geometry/Euclidean_Plane_and_its_Relatives_(Petrunin)/09%3A_Inscribed_angles/9.06%3A_Tangent_half-linesA half-line \([AX)\) is called tangent to the arc \(ABC\) at \(A\) if the line \((AX)\) is tangent to \(\Gamma\), and the points \(X\) and \(B\) lie on the same side of the line \((AC)\). If the arc i...A half-line \([AX)\) is called tangent to the arc \(ABC\) at \(A\) if the line \((AX)\) is tangent to \(\Gamma\), and the points \(X\) and \(B\) lie on the same side of the line \((AC)\). If the arc is formed by a union of two half lines \([AX)\) and \([BY)\) in \((AC)\), then the half-line \([AX)\) is considered to be tangent to the arc at \(A\). Show that there is a unique arc with endpoints at the given points \(A\) and \(C\), that is tangent to the given half line \([AX)\) at \(A\).
- https://math.libretexts.org/Bookshelves/Geometry/Euclidean_Plane_and_its_Relatives_(Petrunin)/17%3A_Projective_Model/17.01%3A_Section_1-Note that the h-plane is mapped to the north hemisphere; that is, to the set of points \((x,y,z)\) in \(\Sigma\) described by the inequality \(z>0\). Suppose that \(\Lambda\) denotes the plane; it con...Note that the h-plane is mapped to the north hemisphere; that is, to the set of points \((x,y,z)\) in \(\Sigma\) described by the inequality \(z>0\). Suppose that \(\Lambda\) denotes the plane; it contains the points \(A\), \(B\), \(P'\), \(\hat P\) and the circle \(\Gamma=\Sigma\cap\Lambda\). (It also contains \(Q'\) and \(\hat{Q}\) but we will not use these points for a while.)
