Contrasting the Euclidean and Taxicab Planes

prepared by:

Joseph Malkevitch
Department of Mathematics and Computing
York College (CUNY)
Jamaica, New York 11451

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Let A = (0, 6), B = (4, 0), and C = (8, 10)

1. Find the Euclidean lengths of the AB, BC, and CA.

2. Find the Taxicab lengths of AB, BC, and CA.

3. Find the midpoint of the each side of the triangle in

a. the Euclidean plane

b. the Taxicab plane

4. a. In the Euclidean plane, find the equations of the perpendicular bisectors of the sides of the triangle. Find the coordinates of the point X (points) where these lines meet. Can you you find the equation of a circle which passes through all of the vertices of the triangle? If there is such a circle what is its radius? What is the Taxicab distance between X and the vertices of the triangle?

b. Determine with respect to Taxicab distance: i. The set of points U which are equidistant from A and B. ii. The set of points V which are equidistant from B and C. iii. The set of point W which are equidistant from points C and A. What can you say about the point (points) that are in common to U, V, and W? Can you find a Taxicab circle which passes through all three of vertices of this triangle? If there is such a circle what are the coordinates of its center and what is the distance of the vertices of the triangle from this center? What is the radius of this circle if it exists?

Comment: In the Euclidean plane a circle which passes through the vertices of a triangle is called the circumcircle of the triangle.

5. Any Euclidean triangle has a circumcircle. Can you describe how to find it?

6. Does every Taxicab triangle have a circumcircle? Can you describe how to find it?

7. Can you find the coordinates of one or more vertices D such that in the Taxicab plane triangle DAB is an equilateral triangle.

Repeat this process for sides BC and CA. Call the the "outwardly built" equilateral triangles on the sides AB, BC, and CA, D, E, and F respectively. Find the lengths of the sides of this this triangle. Does anything noteworthy occur?

Fact: If equilateral triangles are erected outwardly on the sides of an arbitrary triangle in the Euclidean plane, then the vertices of these triangles that are not part of the original triangle form the vertices of an equilateral triangle. This theorem is sometimes called Napoleon's Theorem.


Taxicab Geometry is one of an infinite family of geometries discovered by the mathematician Hermann Minkowski (1864-1909) and sometimes referred to as Minkowski Geometries or Minkowski Planes. Euclidean geometry is also an example of a Minkowski Geometry. The amazing theorem that Minkowski discovered is that if one takes any centrally symmetric convex plane set C then C can serve as the unit circle for a geometry. A set is centrally symmetric if it contains a point O such all the line segments through O meet the set in point for O is the midpoint of the segment formed. (Note that a Euclidean equilateral triangle has three axes of symmetry (mirrors of symmetry) but these mirrors meet at a point which is not a center of symmetry of the equilateral triangle. In fact, a Euclidean equilateral triangle is not centrally symmetric.) Minkowski saw that it was possible to create infinitely many different distance functions for the space of ordered pairs (x, y). To find the distance between two points P and Q using the distance determined by the set C, place a copy of C at P. Now, see how much one must "blow up" (or shrink) C so that it just touches B. This "scaling factor" is the distance between P and Q.