Notes on Euclidean Geometry

Title page
1 Pythagoras Theorem and Its Applications
1.1 Pythagoras Theorem
1.2 Construction of the regular pentagon
1.3 The law of cosines and its applications
2 The circumcircle and the incircle
2.1 The circumcircle and the law of sines
2.2 The incircle and excircles
2.3 Heron\'s formula for the area of a triangle
	2.3.1 Exercise 2.3
3 The shoemaker\'s knife
3.1 The shoemaker\'s knife
3.2 Circles in the shoemaker\'s knife
3.3 Archimedean circles in the shoemaker\'s knife
4 Introduction to Triangle Geometry
4.1 Preliminaries
	4.1.1 Coordinatization of points on a line
	4.1.2 Centers of similitude of two circles
	4.1.3 Tangent circles
	4.1.4 Harmonic division
	4.1.5 Homothety
	4.1.6 The power of a point with respect to a circle
4.2 Menelaus and Ceva theorems
	4.2.1 Menelaus and Ceva Theorems
	4.2.2 Desargues Theorem
	4.2.3 The incircle and the Gergonne point
	4.2.4 The excircles and the Nagel point
4.3 The nine-point circle
	4.3.1 The Euler triangle as a midway triangle
	4.3.2 The orthic triangle as a pedal triangle
	4.3.3 The nine-point circle
4.4 The OI-line
	4.4.1 The homothetic center of the intouch and excentral triangles
	4.4.2 The centers of similitude of the circumcircle and the incircle
	4.4.3 Reflection of I in O
	4.4.4 Orthocenter of intouch triangle
	4.4.5 Centroids of the excentral and intouch triangles
	4.4.6 Mixtilinear incircles
4.5 Euler\'s formula and Steiner\'s porism
	4.5.1 Euler\'s formula
	4.5.2 Steiner\'s porism
5 Homogeneous Barycentric Coordinates
5.1 Barycentric coordinates with reference to a triangle
	5.1.1 Ceva Theorem
	5.1.2 Some basic triangle centers
	5.1.3 The orthocenter
5.2 The inferior and superior triangles
	5.2.1 More triangle centers
5.3 Isotomic conjugates
	5.3.1 Congruent-parallelians point
5.4 Internal center of similitudes of the circumcircle and the incircle