Basic T-Constructions
This activity belongs to the GeoGebra book GeoGebra Principia.
If we fix a point O in the plane, we can consider the taxicab distance from the rest of the points to O.
As we have seen, the points that are T-equidistant from O form a square (with diagonals parallel to the axes). If the radius is r, the perimeter is 8r, so the ratio between the T-circumference and its T-diameter is 4 (instead of ).
By fixing another point I different from O, we establish an orientation O→I and a line. We will take the T-distance from O to I as the unit. We can continue to think of the T-lines as if they were E-lines, as only the way of measuring each segment changes. Remember that pixels force GeoGebra to draw lines composed of horizontal and vertical segments!
Given a point A on the line r, there exists only another point A' on this line at the same distance from O as A. This T-symmetric coincides with the E-symmetric point.
For two distinct points A and B, we can find all the points equidistant from them.
This T-perpendicular bisector does not coincide with the Euclidean perpendicular bisector.
By intersecting the T-perpendicular bisector with the line, we obtain the midpoint, which coincides with the Euclidean midpoint.
Perpendicular and parallel lines are the same as in Euclidean geometry, but the orthogonal projection of a point onto a line does not generally provide the nearest point on the line. (Moreover, "nearest point" is not uniquely determined when the line has a slope of 1 or –1.)
To perform a T-inversion 
, we move points A and I to the horizontal line passing through O, invert (x(A), y(O)) on the dashed E-circle with center O passing through (x(I), y(O)), and create similar triangles that guarantee the new inversion.
The T-inverse of A does not coincide with the E-inverse of A.
, we move points A and I to the horizontal line passing through O, invert (x(A), y(O)) on the dashed E-circle with center O passing through (x(I), y(O)), and create similar triangles that guarantee the new inversion.
The T-inverse of A does not coincide with the E-inverse of A.Author of the construction of GeoGebra: Rafael Losada.