Euclidean Geometry in Mathematical Olympiads（欧几里德几何在数学奥林匹克竞赛）.pdf

C H A P T E R 2 Circles Construct a circle of radius zero. . . Although it is often an intermediate step, angle chasing is usually not enough to solve a problem completely. In this chapter, we develop some other fundamental tools involving circles. 2.1 Orientations of Similar Triangles You probably already know the similarity criterion for triangles. Similar triangles are useful because they let us convert angle information into lengths. This leads to the power of a point theorem, arguably the most common sets of similar triangles. In preparation for the upcoming section, we develop the notion of similar triangles that are similarly oriented and oppositely oriented. Here is how it works. Consider triangles ABC and XY Z. We say they are directly similar, or similar and similarly oriented, if ABC = XY Z, BCA = Y ZX, and CAB = ZXY. We say they are oppositely similar, or similar and oppositely oriented, if ABC = −XY Z, BCA = −Y ZX, and CAB = −ZXY. If they are either directly similar or oppositely similar, then they are similar. We write ABC ∼ XY Z in this case. See Figure 2.1A for an illustration. Two of the angle equalities imply the third, so this is essentially directed AA. Remember to pay attention to the order of the points. T1 T2 T3 Figure 2.1A. T is directly similar to T and oppositely to T . 1 2 3 23 24 2. Circles The upshot of this is that we may continue to use directed angles when proving triangles are similar; we just need to be a little more careful. In any case, as you probably already know, similar triangles also produce ratios