[color=#999999]This activity belongs to the [i]GeoGebra book[/i] [url=https://www.geogebra.org/m/sw2cat9w]GeoGebra Principia[/url].[/color][br][br][br]If we fix a point O in the plane, we can consider the taxicab distance from the rest of the points to O.[br][br]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 [math]\pi[/math]).[br][br]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![br][br]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.[br] [br]For two distinct points A and B, we can find all the points equidistant from them. [br][br] [color=#CC3300]This T-perpendicular bisector does not coincide with the Euclidean perpendicular bisector.[/color][br]  [br]By intersecting the T-perpendicular bisector with the line, we obtain the midpoint, which coincides with the Euclidean midpoint. [br] [br]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.)[br][br]To perform a T-inversion [url=https://en.wikipedia.org/wiki/Inversive_geometry][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url], 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.[br] [br] [color=#CC3300]The T-inverse of A does not coincide with the E-inverse of A.[/color]

[color=#999999]Author of the construction of GeoGebra: [url=https://www.geogebra.org/u/rafael]Rafael Losada[/url].[/color]