Taxicab Betweenness

★ 7/10plane geometry functions Absolute Value Casework

Problem

For points $A(x_1, y_1)$ and $B(x_2, y_2)$ in the coordinate plane, define the taxicab distance

d(A, B) = |x_2 - x_1| + |y_2 - y_1|.

Fix two distinct points $A$ and $B$ .

1. Prove that every point $C(x, y)$ satisfies $d(A, C) + d(C, B) \geqslant d(A, B)$ . 2. Do there exist points $C(x, y)$ with both $d(A, C) + d(C, B) = d(A, B)$ and $d(A, C) = d(C, B)$ ? If so, find all of them; if not, explain why.

Answer

Solution

Difficulty	7/10
Topics	plane geometry, functions, Absolute Value, Casework

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