Here is a text that provides an algebraic approach to plane geometry, making use of the complex numbers.
The Complex Plane and the Euclidean Plane
In outline, here are the contents of the text:
Chapter 1. Basic algebraic results on Euclidean geometry
1.1. The set C of complex numbers
1.2. Metric properties of C
1.3. Rigid motions on C
1.4. Lines and line segments
1.5. Rays, wedges, and angles
1.6. Triangles
1.7. Congruence of triangles
1.8. Isosceles triangles and equilateral triangles
1.9. Similarity
1.10. Squares, rectangles, and other quadrilaterals
1.11. Circles
Chapter 2. Bringing in calculus
2.1. Outline of calculus
2.2. Curves, arclength, and the real measurement of angles
2.3. Exponential and trigonometric functions, and Euler’s formula
2.4. More on Euler’s formula, trigonometric functions, and pi
2.5. Regular polygons
2.6. Area
2.7. Heron’s formula
2.8. Making a trig table
2.9. Euclidean numbers
2.10. Linear fractional transformations
2.11. Argand and the fundamental theorem of algebra
2.12. Planetary orbits: ellipses, parabolas, hyperbolas
2.13. Higher dimensions
2.A. Matrices and their action
The set C of complex numbers arose as an extension of the set R of real numbers in which one could solve the equation z^2 = -1. This was obtained by adding an imaginary unit i, satisfying i^2 = -1. Algebraic operations on R extend to C in a natural fashion, as shown in Section 1.1.
People working with this enlarged number system realized that it is convenient to visualize C as the complex plane, and to utilize the insights of Euclidean geometry. Turning this around, one can perceive that the algebraic structure of C provides an ideal set of tools with which to obtain basic geometric results set out in the books of Euclid. We describe how this works here.
This treatment fills a gap between what students learn about Euclidean geometry in high school and what is needed for more advanced studies. It should be useful to students who have completed a three-semester calculus sequence, and have taken or are currently taking an analysis course that gives a rigorous treatment of the real numbers, the fundamental theorems of calculus, and concepts involving infinite sequences and series of functions.
In Chapter 1 we show that the use of the complex plane C, with its algebraic structure, provides a very pleasant way to develop plane geometry. We have a notion of distance between two points in C, given by d(z,w) = |z-w|, leading to the notion of an isometry F, preserving the distance and having a simple algebraic characterization. We proceed to lines, the notion of parallelism and orthogonality, aided by a special formula for |z+w|^2, involving the inner product, the real part of the product of z and the complex conjugate of w. This in turn leads to the Pythagorean theorem, the cornerstone of Euclidean geometry, and we are on our way, with treatments of triangles, congruences, polygons and circles.
In Chapter 2 the text makes contact with calculus, applies this to a study of arclength, and develops trigonometry in a logical fashion, emphasizing Euler’s formula as a key to that subject. We show how this throws light on classical results, including computations of Archimedes and Ptolemy of perimeters and areas of regular polygons, approximations to pi, and trigonometric tables. Other topics include results on elements of C constructible via compass and straightedge, linear fractional transformations and their application to a three-circle problem of Apollonius, and Argand’s proof of the fundamental theorem of algebra. We include an approach using the complex plane to the planetary orbit problem, whose orbits are given by various conic sections. We close with a brief description of Euclidean geometry in higher dimensions.