Articles of geometric construction

How to construct hyperbolically equidistant points on a line?

In Stillwells’ “Sources of Hyperbolic Geometry ” page 66 figure 3.3 shows an ((incomplete?) construction of hyperbolically equidistant points on a line. I tried to reconstruct the figure but did not manage it can anybody tell me how this figure is constructed? Or another way to construct hyperbolically equidistant points on a line.

Straightedge and compass theory in three dimensions

I’m looking for a reference on the theory of straightedge and compass constructions in three dimensions akin to Euclid’s Elements in two dimensions. More specifically, I mean a theory of geometric constructions where one is allowed lines between any two points, planes through any three non-colinear points, and spheres with a given center and radius. […]

On constructing a triangle given the circumradius, inradius, and altitude .

I was recently pondering about constructing triangles given different attributes of it. I am wondering whether we could construct a triangle given its Circumradius $R$ , Inradius $r$, and length of one altitude.

Construction of a triangle with given angle bisectors

This question already has an answer here: Construct the triangle with given angle bisectors 1 answer

Why can't we construct a square with area equal to a given equilateral triangle?

In modern geometry, given an equilateral triangle, one can’t construct a square with the same area with the use of Hilbert tools. Why is this? The claim seems untrue to me, so there must be something wrong with my understanding. First, given an equilateral triangle of side length $s$, the area of the triangle is […]

Importance of construction of polygons

Why did the ancient Greeks give so much importance to the construction of regular polygons with $n$-sides using only ruler and compass and tried to study for what $n$ was such a construction possible? Until Gauss-Wantzel, this was a famous open problem in Euclidean geometry. Can anyone throw any light on its importance?

Euclidean constructions

I have been playing with a Euclidean geometry application call euclidea. I have been completely stuck on this level for a while now. Any ideas how to solve this problem using only a compass and straight edge?

How to construct a line with only a short ruler

Suppose I want to draw a line between the points A and B but I only have a ruler that covers only something between a fifth and a quarter of the distance between the two points. Also available a compass but with the same limitation. (maximum diameter of a circle is somewhere between a fifth […]

Does there exist a tool to construct a perfect sine wave?

For example, a perfect circle can be constructed using a compass and a perfect ellipse can be constructed using two pins and a piece of string, because a circle can be defined as the locus of points equidistant from a circle point and an ellipse can be defined as the locus of all points such […]

Constructing a circle through a given point, tangent to a given line, and tangent to a given circle

While browsing around about problems similar to the problem of Apollonius, I have found references to constructions of all types of circles. For example, not only is it possible to construct a circle tangent to three given circles, but one can construct a circle through any three points, tangent to any three lines, passing through […]