# Tagged Questions

Questions on the construction of geometrical figures using a limited set of "tools".

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### Prove that there does not exist a $n$-regular polygon $(n\ge 4)$, such that its sides and diagonals are all integers.

Prove that there does not exist a $n$-regular polygon $(n≥4)$, such that its sides and diagonals are all integers. Maybe a famous problem, but I don't know how to solve that.
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### 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 ...
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### Is there a way to draw a 1 degree angle using only ruler and compass?

There are ways to draw 180, 90, 45, 30, 60, ... degree angles. But is there a way to draw a 1 degree angle? In other words how to divide a circle into 360 equal parts?
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### Multiplication of lines of irrational length

Suppose you are given two lines, each has length of an irrational number. How would one, using a straight-edge and compass, draw a line whose length is the product of the two given irrational numbers? ...
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### A straightedge and compass construction: $\left(G,I,Q_a\right)$

Construct $ABC$ with straightedge and compass, given $G,I,Q_a$. $G -$ the intersection point of medians; $I -$ the center inscribed circle; $Q_a -$ point of tangent inscribed circle to the side ...
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### Constructible $n$-gons

Let $\xi$ be the primitive root of unity. If $n=5$, then the minimal polynomial of $xi$ over rationals would have degree $5-1=2^2$ which is a fermat prime so $5$-gon is constructible. If $n=8$, ...
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### Cube root of a line

Well this may be simple but I am not getting it. Give a line segment (of length $l$)(and a segment of unit length if you require) how to construct a line of length $l^{1/3}$ with only a straight ...
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### Construct an equilateral triangle with area equal to a given triangle

It is straightforward to construct (straight-edge and compass) an isosceles triangle with area equal to a given triangle $\triangle ABC$, for instance as follows: Construct the line through $A$ ...
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### Does the collection of algebraic/number-theoretic methods applied to Euclidean Geometry have a name?

I am currently writing an essay on the history of geometry. To educate myself on the subject, I sometimes read the following Wikipedia article on the history of Euclidean Geometry. It seems to me that,...
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### Squaring a line segment

I dont know whether I am being silly or not, but my question is: Given a line segment (say length $l$), how can you draw a line segment of of length $l^2$ using straight edge and compass? I ...
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### Can anyone solve this geometric construction problem?

I remember when I was in high school, one of my all-time favorite books was College Geometry by Nathan Altshiller-Court. Some of its problems kept me wondering for days and even weeks. Now after about ...
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### What makes a geometric construction more or less stable?

As anyone who's actually done geometric construction of n-gons knows, not all construction methods are made equal. Some are very stable (the shape you get is always close to ideal even if you're not ...
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### Is $e$ “constructable” with the appropriate tools?

Of course $e$ cannot be constructed with straightedge and compass. If we allow a marked ruler or a non-rectractable compass (or we use origami...) we can construct numbers like $\sqrt[3]{2}$, but no ...
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### The geometric construction of the $90^\circ, 87^\circ, 3^\circ$ triangle

The construction of the $90^\circ, 45^\circ, 45^\circ$ and the $90^\circ, 60^\circ, 30^\circ$ triangles is well known. How can be constructed a triangle with angles $90^\circ, 87^\circ, 3^\circ$ ...
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### $ax^2+bx+c$ is constructible

If $a$ is constructible, then $\sqrt a$ is constructible. Furthermore, if $a,b,c$ are constructible, then every root of $ax^2+bx+c$ is constructible. I think I know how to prove the first sentence ...
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### Line through a given point

Could someone give a proof of "The line through a given point and parallel to a given line can be drawn". I would be able to do this myself but I am so confused on the wording. As in what are we ...
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### Geometry (Locus and constructions)

I want to find the equation for the locus that is at the same distance from the point $(2,3)$ to the line $x=1$. Im not sure if I am right or wrong? Is the locus just the two point at a distance=1 ...
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### How to construct the circumcenter of a triangle using a compass ONLY.

I just figured out how to find the midpoint between two points using just a compass and no straight edge. A similar approach can be found in this question: Constructing the midpoint of a segment by ...
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### Construction of a triangle using symmetry

I need to construct a triangle $\Delta \textrm{ABC}$ knowing that $t_a = AS$, $|AS| = 6\, cm$, $|\measuredangle \textrm{BCA}| = 30°$ and $|AB| = 5.5 \,cm$. I've been told that it's possible to do it ...
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### What is reflection across parabola?

Reflection across a line is well familiar, reflection across a circle is the inversion, the point at a distance $d$ from the center is reflected into a point on the same ray through the center, but at ...
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### given two concentric circles construct a particular chord

I am stumped by another Euclidea problem - Euclidea problem 9.8: Given two concentric circles $C_1$ and $C_2$ with radians $r_1$ and $r_2$, with $r_1 < r_2 < 2 r_1$ Construct the chord $e$ ...
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### Can we prove that plumb line is vertical to ground?

Using a plumb line to make sure a wall is vertical for instance, is as far as I know one of the most primary tools in the sense that the very-very ancient builders used it as an instrument. I was ...
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### construct triangle given angle and centroid

I am stumped by Euclidea problem 8.11: From a triangle are given angle A and the centroid G Construct the points B and C. Please only a hint
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### Online tool for making Geometric Constructions.

There was a website where it tasked you making different geometric shapes using only a compass and straightedge. I've looked for it and I can't find it or even discussion about it. What I do remember ...
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### Construct a circle passing through a point $X$, which is externally tangent to two given circles

Given two disjoint circles $S_1$ and $S_2$, and a point $X$ external to both of them, is it possible to find the center of a circle that passes through $X$ and touches $S_1$ and $S_2$ tangentially, ...
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### Why is not possible to draw this triangle?

Why is it not possible to draw triangle $DEF$ with $EF=5.5cm$,$\angle E=75^0$ and $DE-DF=1.5cm$?(I used this method for consruction-http://gradestack.com/CBSE-Class-9th-Complete/Construction/...
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### Ruler and compass question

Provide the exact list of steps needed to draw, using ruler and compass, a line $M$ through a given point $A$ and parallel to a given line $L$ (given by two points $B$ and $C$ on it). Assume that $A$ ...
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### Construction of a triangle, given: side, sum of the other sides and angle between them.

Given: $\overline{AB}$, $\overline{AC}+\overline{BC}$ and $\angle C$. Construct the triangle $\triangle ABC$ using rule and compass.
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### Finding square roots of complex number with ruler and compass

Provide the exact list of steps needed to find, with ruler and compass, the two square roots of a given complex number. (The points $0$ and $1$ are given) I don't really understand what I have to ...
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### Construction of major and minor axes of an ellipse given only 2 focus points($F_1,F_2$) and a point $P$ that is on the ellipse.

Construction of major and minor axes of an ellipse given only 2 focus points $(F_1,F_2)$ and a point $P$ that is on the ellipse. Suppose we define $|F_1P|+|PF_2|:=l$ First I constructed the ...
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### Much used compass and straightedge constructions

I am a editor of wikipedia and would like to know which compass and straightedge constructions deserve a place in the list https://en.wikipedia.org/wiki/Compass-and-straightedge_construction#...
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### Accuracy of a Construction

Is there an easy way to find the accuracy of a construction given a straight-edge and compass? For instance setting the point of a compass on an existing line. How do I know how exact that is? Or ...