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

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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 ...
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Can $n$ circles be drawn such that all have a common intersection but no two intersect individually

I was fiddling with plane geometry when a question came into my mind: Can $n$ circles ($n \ge 3$, $n \in \mathbb{N}$) be drawn such that: $C_1 \cap C_2 \cap C_3 \cap \ldots \cap C_n \not = ...
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Straight Edge - Only Geometric Construction

Given a circle, its diameter and a given point on the diameter, find a procedure to construct a line perpendicular to the diameter using only a straight edge. The perpendicular must pass through ...
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Construction using a straight edge only

Given a circle, its diameter and a point on the circle, find a procedure to construct a line perpendicular to the diameter using only a straight edge. The perpendicular must pass through the given ...
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1answer
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You are given two points and a circle. Construct a circle passing through the given two points and tangent to the given circle. [duplicate]

You are given two points and a circle. Construct a circle passing through the given two points and tangent to the given circle. You are allowed to use a straightedge and a compass.
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How to construct a triangle from…?

Two medians and one height (nothing is for the same side!); Outer circle radius, one side and another side's height? Using Compass-and-straightedge construction.
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Can a regular heptagon be constructed using a compass, straightedge, and angle trisector?

Euclid has a magical compass with which he can trisect any angle. Together with a regular compass and a straightedge, can he construct a regular heptagon?
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Constructing the asymptotes of a hyperbola by compass and straightedge.

Is it possible to construct the asymptotes of a hyperbola by compass and straightedge? And if so, how to construct them? I have no idea how to approach the first question. It seems it should be ...
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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 ...
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How can one show algebraically that an angle is constructible?

For example an angle of 30 degrees. I know that geometrically I can obtain the entire 30-60-90 triangle using the standard tools (compass, straightedge and unit length) and by performing iterations. ...
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Trisect unknown angle using pencil, straight edge & compass; Prove validity of technique

This question was posed by my high school geometry teacher, for extra credit: Is it possible, using only a pencil, a straightedge (not a ruler) and a compass to trisect an angle of unknown value? ...
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3answers
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Construct a parallelogram subject to certain conditions

I am having trouble with the following exercise from Dollon and Gilet's Géométrie plane. Two parallel lines $\Delta$ and $\Delta'$ are given, as well as a point $A$ on $\Delta$ and a point $O$ on ...
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1answer
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A straightedge and compass construction: $\left(\widehat{A},r,b-c\right)$

I am looking for an elegant solution of the following problem: Construct $ABC$ with straightedge and compass, given $\widehat{A},r,b-c$. By taking the lines $AB,AC$ as a skew reference system, ...
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4answers
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Construct a triangle given certain lengths related to a bisector

Let $ABC$ be a triangle, and $AD$ the bisector of angle $A$. Write $AB = c$, $AC = b$, $AD = d$, $BD = c'$, $CD = b'$. Using ruler and compass, construct the triangle $ABC$ given the lengths $d$, ...
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2answers
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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 ...
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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 ...
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I think I found a method for Squaring the Circle. But I'm not sure if it's valid.

Here it is: Method for constructing a line of length π: Construct a circle labeled A, with a radius of 1. Bisect circle A. Each of the resulting arcs is now length π. Label one arc B. Align one end ...
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Construct 60° angle through point, other line in only four compass-and-straightedge steps

PROBLEM Here is a surprisingly intriguing challenge posed on Euclidea, a mobile app for Euclidean constructions: Construct a 60° angle through both a point $P$ and an external (infinite) line ...
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Is my observation correct about geometric constructions?

I have observed that it is possible to construct angles which are multiples of 3 with a ruler and a compass (Angles are in degrees). For example, 135°, 45° etc. can be constructed but Angles like 100° ...
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If $tan^2 \theta = \frac{x}{y}$ how can we construct the angle $\theta$?

If we are given the values of $x$ and $y$ and we know that $\tan^2 \theta = \dfrac{x}{y}$ is it possible for us to construct the angle $\theta$?
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Constructible Solutions

We know that if a cubic equation with a rational coefficients has a constructible root, then the equation has a rational root. Now let; $$x^3-2x+2\sqrt{2}=0$$ Could $\sqrt{2}t$ be a viable ...
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Construction of a graph with specific property.

I am trying to find out graphs where eccentricity of every vertex is one except two vertices. And the eccentricity of these two vertices is two. I came to the conclusion that a path graph $P_3$ and a ...
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Predicting Spirals

I am currently in the process of analyzing a polyspiral, a spiral where each successive length drawn is increased at specified increment at the same angle. *Please note the angles selected are the ...
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Are all computable numbers constructable after a countable number of steps?

While looking at another question on this site about constructable numbers I started wondering. If you can take a countable number of steps (possibly infinite) can you draw an interval of a length ...
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Geometric Construction : Construct a Triangle given 3 heights .. [closed]

Given 3 heights : $h_1=5\mathrm{cm}$ ; $h_2=7\mathrm{cm}$ ; $h_3=8\mathrm{cm}$ ... It is required to draw that triangle using only compass and ruler ! N.B.: It is not allowed to calculate the area ...
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1answer
166 views

Construct a triangle with its orthocenter and circumcenter on its incircle.

Construct $\triangle ABC$ such that its orthocenter ($H$) and circumcenter ($O$) are on its incircle. I've tried something by inverting everything WRT circumcircle but don't have proper idea... ...
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How to build a hexagon according to Poincaré model?

Given a side, I know how to build a hexagon in the euclidean geometry. How can i build it in the hyperbolic geometry according to the Poincaré model? By translating every step using hyperbolic circle ...
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A circle can include all but one of n points, but which one can it be?

The answers to the question "Circle enclosing all but one of n points" demonstrate that, given $n$ points, it is possible to construct a circle such that all but one of the points is inside the circle ...
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2answers
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construct triangle given $b-c$, $r$ and $h_{b}$

As in title: the problem is to construct triangle given difference of sides $b$ and $c$, then in-circle radius $r$, and height $h_{b}$. The problem is from a set of problems exercising various ...
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1answer
63 views

Can one construct any n-gon if angle trisection is also allowed?

Suppose one is asked to construct a regular n-gon, but with one extra operation allowed in addition to the standard compass-and-straightedge ones: trisecting any angle. Are all n-gons constructible ...
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1answer
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Finding $\sqrt{17}$ and $\sqrt{257}$ in the regular $17$-gon and $257$-gon?

(Edit: I need to revise this question with my original intent. Pls do not answer it yet. Thanks.) Given the regular $n$-gon formed by the $n$-th roots of unity. For some $n$, how do we find ...
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1answer
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Drawing a circle tangential to 3 circles (internally to one of them)

The two small circles (in black) are equal in radius, and tangential to the large circle. They also touch each other at the center of the large circle. Now, I want to construct a circle (in orange) ...
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Dividing Up A Circular Search Area

BACKSTORY: I need to collect 500 plant samples for strontium analysis. The samples are randomly distributed across a circular area with a radius of 300 kilometers. I have to do this in 30 days, so I ...
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What types of triangles are constructible?

What types of triangles are constructible? I know that equilateral triangles are easily constructed using compass and straightedge, but what about other types of triangles? Can any other ...
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What cube roots are constructible using compass and MARKED straightedge?

DUE DILIGENCE: I have reviewed the list of questions possibly related to the one that I pose below, and I find that none of them address my particular question. Is there a formal definition for the ...
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2answers
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Constructing a line that passes through $P$

I have recently read a book by Heisuke Hironaka. However, the book is not available on English. The book was basically a biography on his life. Heisuke Hironaka says that his high school teacher had ...
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Is this curve defined by an envelope construction known?

Consider the following construction. Start with the standard envelope construction of a cardioid: on a circle, join each point $\theta$ to $2\theta$. Only, instead of joining with a line, join with ...