TY - JOUR

T1 - Existence of special rainbow triangles in weak geometries

AU - Pambuccian, Victor

PY - 2019/1/1

Y1 - 2019/1/1

N2 - We show that, in any ordered plane with a symmetric orthogonality relation which allows for a meaningful definition of acute and obtuse angles, in which all points are colored with three colors, such that each color is used at least once, there must exist both an acute triangle whose vertices have all three colors and an obtuse triangle with the same property. We also show that, in both a geometry endowed with an orthogonality relation, in which there is a reflection in every line, in which all right angles are bisectable, which satisfies Bachmann's Lotschnittaxiom (the perpendiculars raised on the sides of a right angle intersect), and in plane absolute geometry, in which all points are colored with three colors, such that each color is used at least once, there exists a right triangle with all vertices of different colors.

AB - We show that, in any ordered plane with a symmetric orthogonality relation which allows for a meaningful definition of acute and obtuse angles, in which all points are colored with three colors, such that each color is used at least once, there must exist both an acute triangle whose vertices have all three colors and an obtuse triangle with the same property. We also show that, in both a geometry endowed with an orthogonality relation, in which there is a reflection in every line, in which all right angles are bisectable, which satisfies Bachmann's Lotschnittaxiom (the perpendiculars raised on the sides of a right angle intersect), and in plane absolute geometry, in which all points are colored with three colors, such that each color is used at least once, there exists a right triangle with all vertices of different colors.

KW - Acute triangle

KW - axiom system

KW - generalized metric plane

KW - Lotschnittaxiom

KW - obtuse triangle

KW - ordered plane

KW - right triangle

KW - weak geometries

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U2 - 10.1515/gmj-2019-2042

DO - 10.1515/gmj-2019-2042

M3 - Article

AN - SCOPUS:85071156458

SN - 1572-9176

JO - Georgian Mathematical Journal

JF - Georgian Mathematical Journal

ER -