# What is equidistant: Definition and 20 Discussions

A point is said to be equidistant from a set of objects if the distances between that point and each object in the set are equal.In two-dimensional Euclidean geometry, the locus of points equidistant from two given (different) points is their perpendicular bisector. In three dimensions, the locus of points equidistant from two given points is a plane, and generalising further, in n-dimensional space the locus of points equidistant from two points in n-space is an (n−1)-space.
For a triangle the circumcentre is a point equidistant from each of the three vertices. Every non-degenerate triangle has such a point. This result can be generalised to cyclic polygons: the circumcentre is equidistant from each of the vertices. Likewise, the incentre of a triangle or any other tangential polygon is equidistant from the points of tangency of the polygon's sides with the circle. Every point on a perpendicular bisector of the side of a triangle or other polygon is equidistant from the two vertices at the ends of that side. Every point on the bisector of an angle of any polygon is equidistant from the two sides that emanate from that angle.
The center of a rectangle is equidistant from all four vertices, and it is equidistant from two opposite sides and also equidistant from the other two opposite sides. A point on the axis of symmetry of a kite is equidistant between two sides.
The center of a circle is equidistant from every point on the circle. Likewise the center of a sphere is equidistant from every point on the sphere.
A parabola is the set of points in a plane equidistant from a fixed point (the focus) and a fixed line (the directrix), where distance from the directrix is measured along a line perpendicular to the directrix.
In shape analysis, the topological skeleton or medial axis of a shape is a thin version of that shape that is equidistant from its boundaries.
In Euclidean geometry, parallel lines (lines that never intersect) are equidistant in the sense that the distance of any point on one line from the nearest point on the other line is the same for all points.
In hyperbolic geometry the set of points that are equidistant from and on one side of a given line form a hypercycle (which is a curve not a line).

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1. ### Complex mapping z ↦ ω = (z − a)/(z − b)

All points on that line are equidistant from the points a and b. Thus, the length of ##\frac {z - a} {z - b}## is 1, i.e., the points on the unit circle. If the angle of ##z - a## is ##\alpha##, and the angle of ##z - b## is ##\beta##, then the angle of ##\frac {z - a} {z - b}## is ##\alpha -...
2. ### Number of lines equidistant from four points on a plane

Hi, i'm trying to solve this problem. It's exercise 3 on page 5 from this book: Challenging mathematical problem with elementary solutions The solution is on page 41: I'm OK with the 4 circles in case 1: i can pick (inside/outside): ABC + D, ABD + C, ADC + B, BCD + A. What i cannot...
3. ### A Binary fractal tree with equidistant leaves on a circle

Does there exist a binary fractal tree… (reference: http://ecademy.agnesscott.edu/~lriddle/ifs/pythagorean/symbinarytree.htm ) …whose leaves (endpoints) lie on a circle and are equidistant? Consider a binary fractal tree with branches decreasing in length by a scaling factor r (0 < r < 1) for...
4. ### Prove that there is a point equidistant from 4 other points

Homework Statement This problem is taken from S L Loney Coordinate geometry exercise (ch 2)[/B] Prove that a point can be found which is at the same distance from each of the four points ## \bigg(am_1,\dfrac{a}{m_1}\bigg),\bigg(am_2,\dfrac{a}{m_2}\bigg),\bigg(am_3,\dfrac{a}{m_3}\bigg) ## and...
5. ### Net Electric Field for 4 Equidistant Charges

Homework Statement In the figure the four particles form a square of edge length a = 6.10 cm and have charges q1 = 6.35 nC, q2 = -17.9 nC, q3 = 17.9 nC, and q4 = -6.35 nC. What is the magnitude of the net electric field produced by the particles at the square's center? Homework Equations...
6. ### Time for ring of equidistant particles to collapse (gravity)

(I assume that the three section headings below form the template referred to below) 1. Homework Statement n identical equi-distant particles are distributed equi-distantly around the circumference of a ring of radius r in space. Each particles is of mass m, so the total mass of the ring is...
7. ### Equidistant points on the suface of a sphere

If we are asked to place 3 points on the surface of a sphere so that they are equidistant, it's easy to visualize that the result will be such that the three points form an equilateral triangle. If asked to place 4 points it's easy to visualize that the result is such that the points arrange...
8. ### Equation of Parallel Plane Equidistant to a Point

Homework Statement Find the equation of the plane parallel to 2x - y + 2z = 4 if the point P = (3,2,-1) is equidistant from both planes. Homework Equations a(x-x0) + b(y-y0) + c(z-z0) = 0 The Attempt at a Solution I found the normal vector to the given plane first, as this is quite easy...
9. ### Four Equidistant Particles in electric fields

Homework Statement In Fig.22-30, the four particles form a square of edge length a = 6.50 cm and have charges q1 = 7.59 nC, q2 = -10.9 nC, q3 = 11.5 nC, and q4 = -6.06 nC. What is the magnitude of the net electric field produced by the particles at the square's center...
10. ### Lattice Point Equidistant Proof

Hello all, Could someone help me out with this problem? I tried using circle geometries, perpendicular bisectors, and some more pure algebra. Nothing has been "unifying." Here is the problem: Is it possible to have a point Q=(r,s), where r and s are rational, where the point Q is not...
11. ### Equidistant curve of an ellipse

Draw an ellipse. Then draw a curve that is equidistant all the way around the outside of the ellipse. Is that new curve also an ellipse? I've drawn it out and I can show examples where it's obviously not an ellipse, but I can't come up with a good non-visual explanation. I thought that maybe...
12. ### Finding The Equidistant of 2 Points

Homework Statement I need the formula for finding the coordinates of the point on the y-axis that are equildistant from two other pair of points (3,0) and (3,6). Homework Equations i don't know, but these might have something to do with it. \sqrt{} (y2-y1)2+(x2-x1)2 (x2+x1/2...
13. ### Equation for Equidistant Points Between Two Given Points

Homework Statement Find an equation of the set of all points equidistant from the points A(-1,5,3) and B(6,2,-2). Describe the set Homework Equations dot product cross product The Attempt at a Solution the book found the line: 14x - 6y - 10z = 9 which is perpendicular and i guess...
14. ### What Is the Equation for Points Equidistant from Two Given Points in 3D Space?

Homework Statement Find an equation of the set of all points equidistant from the set points A(-1,5,3) and B(6,2,-2). Describe the set. Homework Equations d= sqrt [(x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2] The Attempt at a Solution I solved the distance between A and B and got d=sqrt...
15. ### Equation of Plane- Equidistant with 2 Points

Homework Statement Find the equation of a plane, every point of which is equidistant from the points A(1, 1, 0) and B(5, 3, -2) The Attempt at a Solution I am quite stuck... I wasn't sure if I could find vectors AP and BP and then find their magnitudes using square root x^2 + y^2 + z^2
16. ### Set of points equidistant from two points

Homework Statement Find an equation of the set of all points equidistant from the points A(-1,5,3) and B(6,2,-2). Describe the set. Homework Equations None... The Attempt at a Solution Um, I've drawn a graph of the two points and that's about as far as I could get. I know the...
17. ### Equation of the plane EQUIDISTANT from two points

I got two points that are (2, -1, 1) and (3, 1, 5) and I need to find the equation of the plane equidistant from those points. Anyone?
18. ### How do I find the equidistant point?

1. Find the coordinates of the point on the line y=3x+1 that is equidistant from (0,0) and (-3,4) 2. distance formula 3. I have no idea how to do this. X_X
19. ### Solving the Differential Equation for Curve Passing Through (1,1) with Equidistant Normal

I need help. I am not getting the solution to this: A curve passes through the point (1,1) and has the property that the normal at any point on the curve is equidistant from the x-axis and the origin. Form the differential equation for the curve, find the curve and state which quadrant(s) the...
20. ### What Is the Equation for a Locus Equidistant from the X and Y Axes?

How would you do this question determine the equation of the locus Locus whose points are equidistant from the x and y axis? :cry: I HATE LOCUS!