This classic text explores the geometry of the triangle and the circle, concentrating on extensions of Euclidean theory, and examining in detail many relatively recent theorems. Several hundred theorems and corollaries are formulated and proved completely; numerous others remain unproved, to be used by students as exercises. 1929 edition.
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55 of 55 people found the following review helpful
By J. Bogaarts
Format:Paperback
Recently Dover has reissued two classics on Euclidean geometry, College Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle (Dover Books on Mathematics) and this book. Both books were originally issued in the first half of the 20th century and both were aimed at a college level audience. Both of them also have a considerable amount of so called triangle geometry. As triangle geometry has seen a large upsurge the last years, especially during the last two decennia, there is certainly a need for an English book that gives an overview of the subject including the recent results. These books are useful in this respect but as they are both from the first half of the 20th century, they are out of date. Until a modern treatment of the subject will be available, these two books and the resources on the www will have to do. Altshiller Courts' book has a great set of exercises that can be used as a training ground for geometric problem solving. The problems in Johnsons' book mostly ask for proofs of theorems that are ommited in the text (that's why I give 4 stars). Another drawback of Johnsons' book is that there is no attention paid to geometric constructions. If you are interested in the subject, buy both, its certainly value for money.
The book assumes that you are familiar with simple geometrical concepts like congruence of triangles, parallelograms, circles and the most elementary theorems and constructions as can be found in Kiselev's book Kiselev's Geometry / Book I. Planimetry.
The table of contents:
I Introduction
Prerequisites
Points at infinity
Notation
Directed angles
II Similar Figures
Homothetic figures
Centers of similitude of two circles
Similar figures in general
III Coaxal circles and inversions
The radical axis
Coaxal circles
Inversions
IV Triangles and Polygons
Ratios in the triangle
Quadrangles and quadrilaterals
The theorem of Ptolemy
Triangle and quadrangle theorems
Polygon theorems and exercises
Theorems concerning areas
V Geometry of Circles
The power theorem of Casey
Circles of antisimilitude
Poles and polars
Stereographic projection
VI Tangent Circles
Circles tangent to two circles
Steiner chains; the arbelos
The problem of Apollonius
Four circles touching a circle
VII The theorem of Miquel
The Miquel theorem
Pedal triangles and circles; Simson lines
VII Theorems of Ceva and Menelaos
Theorems of Ceva and Menelaos; applications
Isogonal conjugates
IX Three Notable Points
Fundamental properties of orthocenter and circumcenter
The orthocentric system
Properties of the median point
The polar circle
X Inscribed and Escribed Circles
Fundamental properties
Algebraic formulas; principle of transformation
XI The nine point circle
Properties of the nine point circle
The theorem of Feuerbach
Further properties of Simson lines
XII Symmedian Point and Other Notable Points
Symmedians and the symmedian point
The isogonic centres
Nagel point, Spieker circle, Fuhrmann circle
XIII Triangles in Perspective
The theorem of Desargues
The theorems of Pascal and Brianchon
XIV Pedal Triangles and Circles
Pedal triangles and circles of a quadrangle
Fontené's theorems; the theorem of Feuerbach
The orthopole
XV Shorter Topics
Statical theorems: center of gravity, resultant of vectors
The cyclic quadrangle and its orthocenters
The theorem of Morley
Circles of Droz-Farny
Miscellaneous exercises
XVI The Brocard Configuration
The Brocard points and their properties
The Tucker circles
The Brocard triangles and the Brocard circle
Steiner point and Tarry point
Related triangles
XVII Equibrocardal Triangles
The Neuberg circles
Vertical projection of triangles
Circles of Appolonius and isodynamic points
The circles of Schoute
Generalizations of Brocard geometry
XVIII Three Similar Figures
Similar figures on the sides of a triangle
Three similar figures in general
Index
The book assumes that you are familiar with simple geometrical concepts like congruence of triangles, parallelograms, circles and the most elementary theorems and constructions as can be found in Kiselev's book Kiselev's Geometry / Book I. Planimetry.
The table of contents:
I Introduction
Prerequisites
Points at infinity
Notation
Directed angles
II Similar Figures
Homothetic figures
Centers of similitude of two circles
Similar figures in general
III Coaxal circles and inversions
The radical axis
Coaxal circles
Inversions
IV Triangles and Polygons
Ratios in the triangle
Quadrangles and quadrilaterals
The theorem of Ptolemy
Triangle and quadrangle theorems
Polygon theorems and exercises
Theorems concerning areas
V Geometry of Circles
The power theorem of Casey
Circles of antisimilitude
Poles and polars
Stereographic projection
VI Tangent Circles
Circles tangent to two circles
Steiner chains; the arbelos
The problem of Apollonius
Four circles touching a circle
VII The theorem of Miquel
The Miquel theorem
Pedal triangles and circles; Simson lines
VII Theorems of Ceva and Menelaos
Theorems of Ceva and Menelaos; applications
Isogonal conjugates
IX Three Notable Points
Fundamental properties of orthocenter and circumcenter
The orthocentric system
Properties of the median point
The polar circle
X Inscribed and Escribed Circles
Fundamental properties
Algebraic formulas; principle of transformation
XI The nine point circle
Properties of the nine point circle
The theorem of Feuerbach
Further properties of Simson lines
XII Symmedian Point and Other Notable Points
Symmedians and the symmedian point
The isogonic centres
Nagel point, Spieker circle, Fuhrmann circle
XIII Triangles in Perspective
The theorem of Desargues
The theorems of Pascal and Brianchon
XIV Pedal Triangles and Circles
Pedal triangles and circles of a quadrangle
Fontené's theorems; the theorem of Feuerbach
The orthopole
XV Shorter Topics
Statical theorems: center of gravity, resultant of vectors
The cyclic quadrangle and its orthocenters
The theorem of Morley
Circles of Droz-Farny
Miscellaneous exercises
XVI The Brocard Configuration
The Brocard points and their properties
The Tucker circles
The Brocard triangles and the Brocard circle
Steiner point and Tarry point
Related triangles
XVII Equibrocardal Triangles
The Neuberg circles
Vertical projection of triangles
Circles of Appolonius and isodynamic points
The circles of Schoute
Generalizations of Brocard geometry
XVIII Three Similar Figures
Similar figures on the sides of a triangle
Three similar figures in general
Index
1 of 34 people found the following review helpful
By James J Vail
Format:Paperback|Amazon Verified Purchase
Aristotle Onassis, arrived as a young man,at the New York City port's; He had $.06 in his pocket as he looked at the Statue of Liberty...He said one word ....."Alexander!"....He took the City in a very short time.....Your coffee or sugar might come from one of his Ships!
Euclid was born in Alexandria....300 B.C.-A city that Alexander founded-It is located in Egypt!....Euclid was Alexander's..."Golden Child"...
Euclid Is the Father of Geometry....
Almost every-man Who has Built a building,of any note-on the Earth...Knows Euclid's name....
Euclid dwells in side the Book you sent Me...You don't realize-Thank's!!!!!!!!!!!!!!!!!!!!!!
Euclid was born in Alexandria....300 B.C.-A city that Alexander founded-It is located in Egypt!....Euclid was Alexander's..."Golden Child"...
Euclid Is the Father of Geometry....
Almost every-man Who has Built a building,of any note-on the Earth...Knows Euclid's name....
Euclid dwells in side the Book you sent Me...You don't realize-Thank's!!!!!!!!!!!!!!!!!!!!!!
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