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Triangles > Inscribed circles
In Euclidean geometry, every triangle has a unique inscribed circle. Taxicab geometry, in the usual spirit of a red-headed step-child, does not offer such a broad guarantee.
Regarding inscribed 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 inscribed circles.
The traditional definition of a triangle incircle is the largest circle inscribed within the triangle that is tangent to all three sides of the triangle. Such a circle is guaranteed to exist in Euclidean geometry. As with circumcircles, a taxicab triangle incircle only exists under certain conditions. We begin by formalizing the definition of an incircle in taxicab geometry to clear up concepts like "tangent" which may not transfer precisely.
DEFINITION: An inscribed circle of a taxicab triangle is a taxicab circle entirely contained in a triangle with three of its corners touching the sides of the triangle.
THEOREM (Triangle Incircle Theorem): A triangle has a unique taxicab incircle if and only if it is an inscribed triangle.
Proof: Let triangle ABC be an inscribed triangle. By Inscribed Triangle Theorem 4, the triangle has at least one completely inscribed angle. Without loss of generality, let gamma be an inscribed angle in the triangle at vertex C such that AC and BC have slope between -1 and 1, inclusive (Figure 1). If the triangle has three inscribed angles, choose gamma to be the angle at the intersection of the lines with slope 1 and -1 (see Inscribed Triangle Theorem 3). For the side AB of the triangle opposite gamma, consider a point P with distance from B
The distance from A to P is
The arcs of taxicab circles of radius r_alpha and r_beta centered on A and B, respectively, intersect AB at the same point P and by the Arc Length Theorem both have a length across the interior of the triangle of
Therefore, these arcs form half a taxicab circle inside the triangle. The other half of the taxicab circle remains inside the triangle because the slope of the lines forming the circle are by assumption greater than the slopes of the lines forming the side of the triangle they intersect.
To prove the converse, assume a triangle has a taxicab incircle. If one or zero sides of the circle overlap the triangle (as in Figure 1), both sides of each angle of the triangle are intersected by a line of slope -1 or 1. This implies a line of the same slope passing through the vertex will remain outside the angle making each angle inscribed. If two sides of the circle overlap the triangle, the enclosed angle is by definition completely inscribed. The other two angles are inscribed by the argument above. Therefore, the triangle is an inscribed triangle.
FIGURE 1: Construction of the taxicab incircle of an inscribed triangle.
For a triangle that is not inscribed, an inscribed taxicab circle still exists in a sense. The enclosed circle technically fails to be an incircle by our definition because it does not touch some of the sides of the triangle (see Figure 2). The problem centers on the fact that the triangle is significantly "wider" than it is "tall" and taxicab circles are in a sense not as "flexible" as Euclidean circles.
FIGURE 2: A taxicab circle within a triangle that is not inscribed.
 Thompson, Kevin P. Taxicab Triangle Incircles and Circumcircles (to appear in The Pi Mu Epsilon Journal).
 Sowell, Katye O. Taxicab Geometry - A New Slant, Mathematics Magazine, Vol. 62, No. 4 (Oct 1989), pp. 238-248.
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