The Feuerbach Theorem and Cyclography in Universal Geometry

We have another look at the Feuerbach theorem with a view to extending it in an oriented way to finite fields using the purely algebraic approach of rational trigonometry and universal geometry. Our approach starts with the tangent lines to three rational points on the unit circle, and all subsequent formulas involve the three parameters that define them. Tangency of incircles is treated in the oriented setting via a simplified form of cyclography. Some interesting features of the finite field case are discussed.

The Universal Parabola

We develop classical properties, as well as some novel facts, for the parabola using the more general framework of rational trigonometry. This extends the study of this conic to general fields.

Rational Trigonometry in Higher Dimensions and a Diagonal Rule for $2$-planes in Four-dimensional Space

We extend rational trigonometry to higher dimensions by introducing rational invariants between $k$-subspaces of $n$-dimensional space to give an alternative to the canonical or principal angles studied by Jordan and many others, and their angular variants. We study in particular the cross, spread and det-cross of $2$-subspaces of four-dimensional space, and show that Pythagoras theorem, or the Diagonal Rule, has a natural generalization forsuch $2$-subspaces.

GARIS DI LAPANGAN HIMPUNAN BILANGAN BULAT MODULO 17

n this paper we discuss rational trigonometry in the field F17 ,in particular point, lines and their properties. A unique property in this field is given by the null lines.