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<spacer> <spacer> Angles and Trigonometry > Inscribed Angles

In Euclidean geometry, all angles less than π radians can be represented as an inscribed angle. Since taxicab circles are composed of straight lines, they lack the curvature and "flexibility" of Euclidean circles. This causes some taxicab angles to not be inscribed (Figure 1).


FIGURE 1: A taxicab angle that is not an inscribed angle.

From the examples in Figure 2 it is clear that both the size of an angle and its position affect whether it is inscribed. Some angles just larger than 2 t-radians are not inscribed while other angles just smaller than 4 t-radians are inscribed (for a discussion about t-radians and taxicab angles, see the Angles page).


FIGURE 2: Inscribed taxicab angles. The angles shown are a) strictly positively inscribed, b) completely inscribed, and c) strictly negatively inscribed.

Since the position of an angle plays a role in whether the angle is inscribed, we introduce here a number of definitions for different types of inscribed angles in taxicab geometry.

DEFINITION: A taxicab angle is positively inscribed if a line of slope 1 through its vertex remains outside the angle.

DEFINITION: A taxicab angle is negatively inscribed if a line of slope -1 through its vertex remains outside the angle.

DEFINITION: A taxicab angle is inscribed if it is positively inscribed, negatively inscribed, or both.

DEFINITION: A taxicab angle is completely inscribed if it is both positively inscribed and negatively inscribed.

DEFINITION: A taxicab angle is strictly positively inscribed if it is positively inscribed but not negatively inscribed. Similarly, a taxicab angle is strictly negatively inscribed if it is negatively inscribed but not positively inscribed.

Note that the inscribed angle theorem from Euclidean geometry does not hold in taxicab geometry. Figure 3 illustrates two scenarios where the ratio of the inscribed angle to the central angle subtended by the same arc is different. In the left figure, alpha has measure 1 t-radian and theta has measure 2 t-radians. In the right figure, alpha again has measure 1 t-radian but theta has measure 2.5 t-radians.


FIGURE 3: Failure of the Euclidean inscribed angle theorem in taxicab geometry.

Inscribed angles play a role in the existence of incircles and circumcircles for triangles.

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References
[1] Thompson, Kevin and Tevian Dray. Taxicab Angles and Trigonometry, The Pi Mu Epsilon Journal, Worcester, MA. Vol. 11, No. 2 (Spring 2000), pp. 87-96.
[2] Thompson, Kevin P. Taxicab Triangle Incircles and Circumcircles (to appear in The Pi Mu Epsilon Journal).
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