Lines, Planes, and Perspective

Aside from points, as we have discussed in the previous post, the other two undefined terms in Geometry are lines and planes. A line may be drawn through two points, while three points are needed to determine a plane. The representations of these undefined terms are the building blocks of Euclidean Geometry.  They can be combined to create shapes, drawings, and sketches such as the painting shown in the first figure. Looking at the painting makes us realize that almost all the things around us are  mostly basic geometric shapes.

Bruce Cohen

In the painting above, we can easily see geometric shapes such as rectangles, triangles, trapezoids , and parallelograms. We can also see curves and arcs in vases, flowers, and fruits.  Notice that although the painting seems to be only made by these shapes, the artist has made it look very realistic. For example, the window frames located at the left side of the painting are of the same size, but the artist made the ‘nearer’ frame larger to give a somewhat three dimensional effect.  In doing so, the painter considered the distance of the window frames from the observer.  The farther the frame, the smaller its size. Observe that this technique is more apparent in the painting by Vincent Van Gogh in the second figure. » Read more

Properties of Similar Triangles Part 2

This is the third and the conclusion of the Triangle Similarity Series. The two prequels  are 1. Introduction to Similarity and 2. Properties of Similar Triangles (Part 1).


In the previous post, we have investigated the properties of similar triangles. We have learned that corresponding angles of similar triangles are congruent. In this post, we are going to discuss more about the properties of similar triangles.  If you have not performed the investigation in the previous post, you can use the applet below.

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You would have realized from your exploration of the applet that aside from the angles, there is also something unique about the side lengths of the corresponding sides of the triangles (check the Show/Hide Side Length check box above).  We can verify they have the same ratio.  That is, if triangle ABC is similar to triangle DEF, then the following relationships hold: » Read more