Sheet E (Geometric Structures)

**Equilateral Triangles**

prepared by:

Joseph Malkevitch

Department of Mathematics and Computing

York College (CUNY)

Jamaica, New York 11451

Email: malkevitch@york.cuny.edu

web page:

http://www.york.cuny.edu/~malk

Definition: A triangle is called equilateral if the lengths of all three of its sides are the same.

1. a. Find the possible vertices of an equilateral triangle in the euclidean plane with one side having the endpoints (0, 0) and (2, 0).

b. Find the possible vertices of an equilateral triangle in the taxicab plane with one side having the endpoints (0, 0) and (2, 0).

2. a. Find the possible vertices of an equilateral triangle in the euclidean plane with one side having the endpoints (0, 0) and (1, 1).

(Hint: One approach would be to write down the equations of circles centered at (0,0) and (1, 1) with a radius of 2. This approach imitates the synthetic approach that Euclid gives for dealing with this question.)

b. Find the possible vertices of an equilateral triangle in the taxicab plane with one side having the endpoints (0, 0) and (1, 1).

(Hint: Find the set of points which are equidistant (taxicab distance) from (0, 0) and (1, 1). Now one wants to find those points in this set which are two units from (0, 0) and (1, 1).)

3. Discuss the set of points that can be common to two euclidean circles of the same radius. Discuss the set of points that can be common to two taxicab circles of the same radius.