Construct in the Cartesian coordinate system two triangles t

Construct, in the Cartesian coordinate system, two triangles that are congruent in the taxicab sense. Make a conjecture about the minimal conditions for triangle s congruence in Taxicab Geometry. (You do not need to prove your conjecture).In Euclidean geometry, for any given circle and any given slope, there exists a line V that is tangent to the circle with that slope. Is this true in Taxicab Geometry? Why or/why not? Explain why, for any line m, the composition of r_m with itself is the identity mapping. In other words, explain why r_m r_m (P) = P for all points P.

Solution

1) Consider the triangle ABC with A(0,0), B(0,2) and C(2,0). It has sides 2,2 and 2 and rt angled at C

Consider the traingle PQR with P(9,0), Q(2,0) and R(1,-20.

These two triangels satisfy the classic SAS conditions, but are not congruent.

The minimal requirements for two triangles in Taxicab metric is the SASAS (side angle side angle side) condition.

(2) Consider the \"unit circle \" of points in the plane with centre (0,0) and radius 1. This is composed of the four lines |x|+|y|=1 and there are only four possible directions for tangents.

So the statement is false.

(3) r_m² is the reflection of the reflection wrt the line m. So every point is first taken to its image and its image is taken to the point by r_m. Hence r_m² = Identity map