# Construction ruler and compass [logic]

Mathematics Asked by Meulu Elisson on November 6, 2020

Carl is given three distinct non-parallel lines $$ell_1, ell_2, ell_3$$ and a circle $$omega$$ in the plane. In addition to a normal straightedge, Carl has a special straightedge which, given a line $$ell$$ and a point $$P$$, constructs a new line passing through $$P$$ parallel to $$ell$$. (Carl does not have a compass.) Show that Carl can construct a triangle with circumcircle $$omega$$ whose sides are parallel to $$ell_1,ell_2,ell_3$$ in some order.

Attempt: To build a triangle with a circle, just have $$3$$ points in that circle, at each point draw a line tangent to the circle and parallel to one of the given lines.

Am I right?

Alright, the Poncelet-Steiner theorem says we can do anything a compass and straightedge can do, if only we can construct the center of the given circle. That turns out to be fairly easy. Given any line, we can construct a diameter of the fixed circle that is perpendicular to the line. Now that I think of it (it's not in the diagram) we might as well make that first diameter, then simply make the diameter perpendicular to that. The intersection of the two (perpendicular) diameters is the center of the circle.

See attached diagram. The fixed circle is in red, so is the initial line being used.

Given two diameters that are not identical, the center is just where the diameters intersect.

So, the original problem ( inscribed triangle) can be solved. We know that by Correct answer by Will Jagy on November 6, 2020

## Related Questions

4  Asked on November 26, 2021

### Bijection between tensors and permutations (in linear $O(n)$ time)

1  Asked on November 26, 2021 by nikos-m

### How to Solve a Stars and Bars Discrete Math Problem

2  Asked on November 26, 2021 by kristen-m-day

### Closed form of $int_0^infty arctan^2 left (frac{2x}{1 + x^2} right ) , dx$

3  Asked on November 26, 2021

### Finding the limit of $mathbb{E}[theta^n]/mathbb{E}[theta^{n-1}]$

1  Asked on November 26, 2021

### Degree of a determinant

1  Asked on November 26, 2021

### Are basic feasible solutions, vertices, and extreme points equivalent for semidefinite programs (SDPs)?

1  Asked on November 26, 2021

### Find all functions $f:mathbb{R}^+to mathbb{R}$ such that $xf(xf(x)-4)-1=4x$

2  Asked on November 26, 2021

### Fourier transform of $1/ sqrt{m^2+p_1^2+p_2^2+p_3^2}$

1  Asked on November 26, 2021 by sebastien-b

### Optimisation of norm of matrices without the elements on diagonals

0  Asked on November 26, 2021 by nikowielopolski

### Given $frac{z_1}{2z_2}+frac{2z_2}{z_1} = i$ and $0, z_1, z_2$ form two triangles with $A, B$ the least angles of each. Find $cot A +cot B$

2  Asked on November 26, 2021 by seo

### Notation for “and”

1  Asked on November 26, 2021

### Proving that, given a surjective linear map, a set is a generator of a vector space

1  Asked on November 26, 2021 by jd_pm

### Production function problem (Lagrange multiplier)

1  Asked on November 26, 2021

### Brezis-Kato regularity argument – Some questions about Struwe’s proof Part II

1  Asked on November 26, 2021 by danilo-gregorin-afonso

### Understanding the multidimensional chain rule

1  Asked on November 26, 2021 by atw

### Prove identity matrix with singular value decomposition

1  Asked on November 26, 2021 by hojas

### Number of possible n-towers

1  Asked on November 26, 2021 by aatish-five

### Convergence of $sqrt{1-sqrt{{1}-sqrt{{1}-sqrt{{1-sqrt{{1}}}…}}}}$

1  Asked on November 24, 2021

### Is this series $sum_{n=0}^{infty} 2^{(-1)^n – n} = 3$ or $approx 3$

2  Asked on November 24, 2021