Reference request: Elementary geometry done by Geometric Algebra

Question

I'm impressed by geometric algebra (aka Clifford algebra, done in the style of Hestenes et al), but feel unfamiliar compared to analytical geometry, due to not having done many exercises in it.

I ask for recommendations to material where geometric algebra is applied to elementary geometry problems, such as the incircles of triangles, Desargue's Theorem, circle inversions, etc.

As examples, New foundations for classical mechanics (Hestenes 2012) has some elementary geometry. Conic Sections and Meet Intersections in Geometric Algebra (Hitzer, 2013) treats conic sections, but it's just a paper rather than a full book.

Answer 1

Forder, Calculus Of Extension (1941) is easily accessed online and is worth a browse.

It is based on the Grassmann (aka exterior algebra) part of geometric algebra, and at least touches on the topics you mention, in addition to the topic of circles and systems of circles. But it often presupposes you have some existing knowledge of geometry, so it is not a ground-up development of the topic.

I can't resist adding a quotation I found in Smith, Using Geometric Algebra: A High-School-Level Demonstration of the Constant-Angle Theorem:

[T]raditional geometry is our heritage and we must preserve it. Traditional geometry is an expression of pure human genius and its study has a very favorable effect on brain development (in the light of modern knowledge about neuroplasticity). On the other hand, geometric algebra, regardless of its power, is not a substitute for everything, and especially it is not a “magic wand” or some “royal path” into geometry. Geometric algebra simply introduces clarity into the question of vector multiplication, with far-reaching consequences. However, caution is required: (in my opinion) you can’t be good at GA if you skip Euclid.

[ - Miroslav Josipovic]

Update: One attempt to go through the basics of plane geometry using geometric algebra is Plane Geometry Through Geometric Algebra (Calvet). It includes Desargues' Theorem and inversion.