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Kevin's Corner

<spacer> <spacer> Triangles > Inscribed Triangles

Triangles in taxicab geometry with one or more inscribed angles have special properties. A number of these will be needed in our investigation of taxicab triangle circumcircles and incircles.

THEOREM 1: If a triangle contains a strictly positively inscribed angle, then the other angles are negatively inscribed.

Proof: If an angle alpha of a triangle is strictly positively inscribed, then for a neighboring angle to not be negatively inscribed it would have to be larger than (4 - alpha) t-radians (see Figure 1) thus violating the angle requirements of a triangle in taxicab geometry (see Basic Triangle Theorem 1).


FIGURE 1: The neighboring angles of a strictly positively inscribed angle of a triangle must be negatively inscribed.

COROLLARY 2: If a triangle contains a strictly negatively inscribed angle, then the other angles are positively inscribed.

A triangle with a completely inscribed angle can only take on certain shapes and positions. Adding additional completely inscribed angles results in a very small class of triangles as the following theorem illustrates.

THEOREM 3: A triangle has three completely inscribed angles if and only if two of its sides have slope -1 or slope 1.

Proof: Suppose alpha and beta are completely inscribed angles at vertices A and B of a triangle. Without loss of generality, suppose alpha is inscribed such that AB and AC have slope between -1 and 1 with AB having a slope strictly less than 1 (Figure 2). Since beta is also completely inscribed, the slope of BC is between -1 and 1, inclusive. If the slope of BC is strictly less than 1, then there exists a line of slope 1 through C passing through the angle gamma making it not positively inscribed. Similarly, if the slope of AC is strictly greater than -1, then there exists a line of slope -1 through C passing through the angle gamma making it not negatively inscribed. Therefore, the triangle has three completely inscribed angles if and only if the rays of angle gamma have slope 1 and -1.


FIGURE 2: Defining a triangle with three completely inscribed angles.

Inscribed Triangles

Triangles whose angles are all inscribed will be of great interest to us. Having a term for this type of triangle will be useful and, in the end, meaningful. While it may seem presumptuous, calling such a triangle an inscribed triangle is fully justified by the results for triangle circumcircles and incircles.

DEFINITION: A triangle is inscribed if all of its angles are inscribed.

Exploring some properties of inscribed triangles will aid in our study of taxicab triangle circumcircles and incircles. There are minimal properties that an inscribed triangle must have. In fact, by imposing the restriction that all three angles must be inscribed, a more stringent condition results. The following theorem illustrates one of the more fundamental and useful properties of an inscribed triangle.

THEOREM 4: An inscribed triangle has at least one completely inscribed angle.

Proof: By way of contradiction and without loss of generality, assume a triangle has a strictly positively inscribed angle alpha. By Theorem 1 the neighboring angles must be negatively inscribed. Suppose one of these angles beta is strictly negatively inscribed. Then by Corollary 2, the other angles must be positively inscribed. So, gamma is completely inscribed (Figure 3). Therefore, an inscribed triangle cannot be constructed with zero completely inscribed angles.


FIGURE 3: An inscribed triangle must have at least one completely inscribed angle.

The following theorem is an extension and generalization of Theorem 1 where a hint of alternating inscribed angle properties among the angles of a triangle was seen. Theorem 4 above provides the flexibility in obtaining the following result.

THEOREM 5: Neighboring angles of an inscribed triangle have opposite inscribed angle properties (completely inscribed angles may be selected to be either positively or negatively inscribed as needed).

Proof: If an angle of a triangle is strictly positively inscribed, then by Theorem 1 the other angles are negatively inscribed. By Theorem 4 one of these angles must be completely inscribed. Therefore, neighboring angles in this triangle have opposite inscribed angle properties. The result is similar when beginning with a strictly negatively inscribed angle.

The only remaining case is a triangle with only completely inscribed angles. The result follows immediately.

The alternating property in an inscribed triangle is key to the construction of taxicab circumcircles for triangles.

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References
[1] Thompson, Kevin P. Taxicab Triangle Incircles and Circumcircles (to appear in The Pi Mu Epsilon Journal).
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