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Kevin's Corner

<spacer> <spacer> Triangles > Circumscribed circles

In Euclidean geometry, every triangle has a unique circumscribed circle. In taxicab geometry there are limitations on which triangles have circumcircles.

Regarding circumscribed circles, the foundational difference between Euclidean geometry and taxicab geometry is the nature of inscribed angles. Since not all angles are inscribed, there exist triangles that are not inscribed triangles. We will see that this causes not all taxicab triangles to have circumscribed circles.

The traditional definition of a triangle circumcircle, or circumscribed circle, is a circle that passes through the vertex points of the triangle. In Euclidean geometry, all triangles have a unique circumcircle. In taxicab geometry, the existence of circumcircles is more restrictive.

THEOREM 1 (Triangle Circumcircle Theorem): A triangle has a taxicab circumcircle if and only if it is an inscribed triangle.

Proof: If a taxicab circumcircle passes through the vertices of a triangle and completely encloses the triangle, by definition the angles of the triangle are inscribed since a taxicab circle is composed of lines of slope 1 and -1.

To prove the converse, suppose the triangle is an inscribed triangle. By Inscribed Triangle Theorem 5, neighboring angles of the triangle have opposite inscribed angle properties. Therefore, we may select three lines of slope 1 or -1 through the vertices of the triangle in an alternating fashion to construct three sides of a taxicab circle around the triangle. Since at least one angle of the triangle is completely inscribed, this can be done in multiple ways. At the completely inscribed angle, a line should be selected that maximizes the side of the circle enclosed by the 3 lines (the solid lines of Figure 1, with the dotted lines showing the alternate choice). By construction, the fourth side of the taxicab circle will enclose the triangle.


FIGURE 1: Construction of the taxicab circumcircle of an inscribed triangle.

The Triangle Circumcircle Theorem justifies our name for inscribed triangles. This type of triangle is precisely the kind for which circumscribed circles exist. From this result, we can easily obtain the taxicab version of the Euclidean geometry "three points define a unique circle" theorem.

COROLLARY 2 (Three-point Circle Theorem): Three non-collinear points lie on a taxicab circle if and only if the triangle they form is inscribed.

So, in contrast to Euclidean geometry, three non-collinear points do not always lie on a circle in taxicab geometry. If the triangle formed by the points contains an angle that is not inscribed, then no taxicab circle can be drawn passing through the points.

It should also be carefully noted that taxicab triangle circumcircles are not necessarily unique. (This means three non-collinear points do not necessarily lie on a unique taxicab circle.) All non-unique cases occur when one or more sides of the circumcircle pass through two vertices. This often creates flexibility in defining possible circumcircles. Three cases are shown in Figure 2. In the first example, exactly one side has a slope of 1 or -1 and the other sides are of equal taxicab length and longer than the first side (noting that the triangle does not necessarily need to be acute). In this case a circumcircle can be shifted diagonally within a limited range around the triangle. For the second example, one side has slope 1 or -1 and another side is parallel to an axis. Larger and larger circles encompass this type of triangle. These circles also encompass the third example along with another set of circles in a limited range of movement as in the first example. In this case the triangle is isoceles (but not equilateral) with two sides having slope 1 or -1.


FIGURE 2: Triangles with infinite circumcircles.

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References
[1] Thompson, Kevin P. Taxicab Triangle Incircles and Circumcircles (to appear in The Pi Mu Epsilon Journal).
[2] Çolakoğlu, Harun Bariş and Rüstem Kaya. A Synthetic Approach to the Taxicab Circles, Applied Sciences, Vol. 9 (2007), pp. 67-77.
[3] Tiana, Songlin; Shing-Seung Soa; and Guanghui Chen. Concerning Circles in Taxicab Geometry, International Journal of Mathematical Education in Science and Technology, Vol. 28 (1997), pp. 727-733.
[4] Sowell, Katye O. Taxicab Geometry - A New Slant, Mathematics Magazine, Vol. 62, No. 4 (Oct 1989), pp. 238-248.
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