Enjoyment of Euclidean planar triangles

Abstract

This article adopts the following classification for a Euclidean planar , purely based on angles alone. A Euclidean planar triangle is said to be acute angled if all the three angles of the Euclidean planar are acute angles. It is said to be right angled at a specific vertex, say B, if the angle is a right angle with the two remaining angles as acute angles. It is said to be obtuse angled at the vertex B if is an obtuse angle, with the two remaining angles as acute angles. In spite of the availability of numerous text books that contain our human knowledge of Euclidean plane geometry, softwares can offer newer insights about the characterizations of planar geometrical objects. The author's characterizations of triangles involve points like the centroid G, the orthocentre H of the , the circumcentre S of the , the centre N of the nine-point circle of the . Also the radical centre rc of three involved diameter circles of the sides BC, AC and AB of the provides a reformulation of the orthocentre, resulting in an interesting theorem, dubbed by the author as ‘Three Circles Theorem’. This provides a special result for a right-angled , again dubbed by the author as ‘The Four Circles Theorem’. Apart from providing various inter connections between the geometrical points, the relationships between shapes of the triangle and the behaviour of the points are reasonably explored in this article. Most of these results will be useful to students that take courses in Euclidean Geometry at the college level and the high school level. This article will be useful to teachers in mathematics at the high school level and the college level.

Keywords:

• Pythagorean theorem
• circumcircle
• nine-point circle
• the orthocentre
• radical axes
• radical centre

AMS Subject Classification:

• 11R04

Acknowledgements

The author would like to thank the referees for their valuable suggestions.