Question 88
For two points $A = (x_1, y_1)$ and $B = (x_2, y_2)$ in $\mathbb{R}^2$, their distance can be defined as,
$$
d(A, B) = |x_1 - x_2| + |y_1 - y_2|.
$$
a) The perpendicular bisector of two points in the plane is the locus of points equally spaced from those two points. Determine the perpendicular bisector of two arbitrary points according to the new definition.
b) The Apollonius circle of two points in the plane is the locus of points for which the ratio of their distances to those two points is equal to a constant value $m$ ($m \neq 1$). Determine the Apollonius circle of two arbitrary points according to the new definition.
c) The distance from point $A$ to line $l$ is defined as
$$
\min\{d(A, P) | P \in l\}.
$$
In Euclidean Geometry, a parabola in the plane is the locus of points equally distanced from a point and a line. Determine how a parabola will look like according to the new definition.
d) Let $ABC$ be a triangle in the plane, and $X_a$ a point on side $BC$ such that,
$$
d(A, B) + d(B, X_a) = d(A, C) + d(C, X_a).
$$
$X_b$ and $X_c$ are defined similarly. Are $AX_a$, $BX_b$ and $CX_c$ concurrent?
e) Is it possible to have an infinite number of points in the plane such that their mutual distances are squares of natural numbers?
$$
d(A, B) = |x_1 - x_2| + |y_1 - y_2|.
$$
a) The perpendicular bisector of two points in the plane is the locus of points equally spaced from those two points. Determine the perpendicular bisector of two arbitrary points according to the new definition.
b) The Apollonius circle of two points in the plane is the locus of points for which the ratio of their distances to those two points is equal to a constant value $m$ ($m \neq 1$). Determine the Apollonius circle of two arbitrary points according to the new definition.
c) The distance from point $A$ to line $l$ is defined as
$$
\min\{d(A, P) | P \in l\}.
$$
In Euclidean Geometry, a parabola in the plane is the locus of points equally distanced from a point and a line. Determine how a parabola will look like according to the new definition.
d) Let $ABC$ be a triangle in the plane, and $X_a$ a point on side $BC$ such that,
$$
d(A, B) + d(B, X_a) = d(A, C) + d(C, X_a).
$$
$X_b$ and $X_c$ are defined similarly. Are $AX_a$, $BX_b$ and $CX_c$ concurrent?
e) Is it possible to have an infinite number of points in the plane such that their mutual distances are squares of natural numbers?