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One of the most famous texts in the history of mathematics is the ancient Chinese Nine chapters. In this article we describe its contents and impact, but also comment it from a contemporary perspective. [ABSTRACT FROM AUTHOR]

Using the theory of elliptic curve, we show that all right triangles, such that the sum of the area and the square of the sum of legs is a square, are given by an infinite set. Similarly, we get all right triangles such that the sum of the area and the square of the semi-perimeter is a square. Using the theory of Pell's equation, we prove that there are infinitely many non-primitive right triangles such that the sum of the area and the hypotenuse (or the smaller leg) is a square, and an infinity of primitive right triangles such that the sum of the area and the smaller leg (or the perimeter, the semi-perimeter, the larger leg) is a square. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Given a triangle, we shall construct a circle which is tangent to both the incircle and the circumcircle at a vertex of the triangle. This construction can be considered as the generalization of Feuerbach's Theorem on right triangle. [ABSTRACT FROM AUTHOR]

By the theory of elliptic curves, we show that there are infinitely many integer right triangle and integer parallelogram pairs with a common area and a common perimeter. [ABSTRACT FROM AUTHOR]

The concept of angles is one of the foundational concepts to develop of geometric knowledge, but it remains a difficult concept for students and teachers to grasp. Exiting studies claimed that students' difficulties in learning of the concept of angles are based on learning of the multiple definitions of an angle, describing angles measuring the size of angles, and conceiving different types of angles such as 0-line angles, 1- line angles, and 2-line angles. This study was designed to gain better insight into pre-service secondary mathematics teachers' (PSMTs) mental constructions of the concept of angles from the perspective of Action-Process- Object-Schema (APOS) learning theory. The study also explains what kind of mental constructions of angles is needed in the right triangle context. The four PSMTs were chosen from two courses at a large public university in the Midwest United States. Using Clements' (2000) clinical interview methodology, this study utilized three explanatory interviews to gather evidence of PSMTs' mental constructions of angles and angle measurement. All of the interview data was analyzed using the APOS framework. Consistent with the existing studies, it was found that all PSMTs had a schema for 2-line angles and angle measurement. PSMTs were also less flexible on constructions of 1-line and 0-line angles and angle measurement as it applied to these angles. Additionally, it was also found that although PSMTs do not have a full schema regarding 0-line and 1-line angles and angle measurement, their mental constructions of 1-line and 0-line angles and angle measurement were not required in right triangles, and the schema level for 2- line angles was sufficient for constructions of right triangle context. [ABSTRACT FROM AUTHOR]

We say that a triangle $$T$$ tiles the polygon $$\mathcal A $$ if $$\mathcal A $$ can be decomposed into finitely many non-overlapping triangles similar to $$T$$ . A tiling is called regular if there are two angles of the triangles, say $$\alpha $$ and $$\beta $$ , such that at each vertex $$V$$ of the tiling the number of triangles having $$V$$ as a vertex and having angle $$\alpha $$ at $$V$$ is the same as the number of triangles having angle $$\beta $$ at $$V$$ . Otherwise the tiling is called irregular. Let $$\mathcal P (\delta )$$ be a parallelogram with acute angle $$\delta $$ . In this paper we prove that if the parallelogram $$\mathcal P (\delta )$$ is tiled with similar triangles of angles $$(\alpha , \beta , \pi /2)$$ , then $$(\alpha , \beta )=(\delta , \pi /2-\delta )$$ or $$(\alpha , \beta )=(\delta /2, \pi /2-\delta /2)$$ , and if the tiling is regular, then only the first case can occur. [ABSTRACT FROM AUTHOR]