Euclidian geometry conjures up images of shapes with all of their angles, lines, tick marks, and sides. The most picturesque branch of high school mathematics is geometry. Geometry after all is the science or art of measuring the earth. It started with the images of the created world around us and is often learned using tools that help us transition the shapes of the world onto a two-dimensional surface. But this was not always the case. In fact, the idea of "a picture is worth a thousand words" was fairly foreign to Euclid. The father of Geometry explains all of his theorems and postulates with words. Yes, he uses a compass and straightedge and tell us to use them as well, but written verbal language was the only tool available to him to explain how to construct. In geometry we are exposed to the ancient world by studying Euclid's elements and the language he used to explain these truths. Reading Euclid is hard, but it also helps grow our mind to think more like the master geometer as we become fellows in a great tradition.
Modern geometry students can rejoice in their understanding of the language of algebra and the efficient use of variables, ratios, and operators to explain the truth in the picture. If you think of it, geometry is a study of multiple layers: the algebra describes the picture which describes the reality of the world. Layers upon layers of truth nestled together in one fantastic world.