Passage

For points $P\equiv \left({x}{1},{y}{1}\right)$ and $Q\equiv\left({x}{2},{y}{2}\right)$ of the coordinate plane, a new distance $d\left(P,Q\right)=\left|{x}{1}-{x}{2}\right|+\left|{y}{1}-{y}{2}\right|$ Let $O\equiv\left(0,0\right), A\equiv\left(2,3\right)$ and $C\equiv\left(4,3\right)$ are four fixed points on the $x-y$ plane. On the basis of the above information, answer the following questions:

Distance between the circumcentre and orthocentre of $\triangle ABC$ is

Using the defined metric, the Manhattan distance between A(2, 3) and C(4, 3) is the sum of the absolute differences of their coordinates, giving |2 - 4| + |3 - 3| equals 2. Because the segment AC is horizontal, its perpendicular bisector is the vertical line x equals 3. To form a triangle ABC with a circumcentre and orthocentre, the third vertex B must lie on the line x equals 3 (for example, at (3, 6) to form an isosceles right triangle), which forces the circumcentre to the midpoint of the base at (3, 3) and the orthocentre to the right-angle vertex B at (3, 10). The metric distance between (3, 3) and (3, 10) is |3 - 3| + |3 - 10| equals 7.

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