Questions tagged [plane-geometry]

Plane geometry is a subfield of Euclidean geometry, restricted to the flat two-dimensional space. Plane geometry studies shapes, ratios and relative locations of 2D figures which can be embedded into 2D plane.

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The most effective windshield-wiper setup. (Packing a square with sectors)

I was on the bus on the way to uni this morning and it was raining quite heavily. I was sitting right up near the front where I could see the window wipers doing their thing. It made me think "what is the best configuration of window wipers for…
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Why are turns not used as the default angle measure?

Why is $2\pi$ radians not replaced by $1$ turn in formulas? The majority of them would be simpler. If such a replacement was proposed earlier, why was it declined?
DSblizzard
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$\triangle ABC$ with a point $D$ inside has $\angle BAD=114^\circ$, $\angle DAC=6^\circ$, $\angle ACD=12^\circ$, and $\angle DCB=18^\circ$.

Let $ABC$ be a triangle with a point $D$ inside. Suppose that $\angle BAD=114^\circ$, $\angle DAC=6^\circ$, $\angle ACD=12^\circ$ and $\angle DCB=18^\circ$. Show that $$\frac{BD}{AB}=\sqrt2.$$ I am requesting a geometric proof (with as little…
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Two individuals are walking around a cylindrical tower. What is the probability that they can see each other?

It'd be of the greatest interest to have not only a rigorous solution, but also an intuitive insight onto this simple yet very difficult problem: Let there exist some tower which has the shape of a cylinder and whose radius is A. Further, let…
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$a^x+b^x=c^x$ in geometry

The Pythagorean theorem. Let $A$, $C$, $B$ be three points on a line in this order, and let $D$ be another point, such that $\angle ADC =\angle CDB = 60^\circ$. Let $a=AD$, $b=BD$, $c=CD$. Then, $$a^{-1} + b^{-1} = c^{-1}.$$ Let $C_1$, $C_2$, $C_3$…
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Tiling the plane with consecutive squares

For which $n$ is it possible to find a region $R$ made of non-overlapping squares of side length $1,2,\ldots,n$ which tiles the plane? $n=1$ is trivial, and $n=2$ works as well. However, for $n\geq3,$ I am unable to find $R$ that work. Obviously, we…
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If two convex polygons tile the plane, how many sides can one of them have?

The set of convex polygons which tile the plane is, as of $2017$, known: it consists of all triangles, all quadrilaterals, $15$ families of pentagons, and three families of hexagons. Euler's formula rules out strictly convex $n$-gons with $n\ge 7$.…
RavenclawPrefect
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Find angle UFO in the picture attached

I sent this problem to Presh Talwalkar who suggested me to send it to this site. I tried many things but was not able to find the correct solution. I made various segments trying to get an equilateral triangle similar to the Russian triangle…
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Does $\mathbb{R}^2$ Contain Uncountably Many Disjoint Copies of the Warsaw Circle?

The Warsaw Circle is defined as the closed topologist's sine curve, with an additional arc attached at its free end point and one of the end points of the critical line: Since we don't have an uncountable collection of disjoint open sets in the…
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Trisect a quadrilateral into a $9$-grid; the middle has $1/9$ the area

Trisect sides of a quadrilateral and connect the points to have nine quadrilaterals, as can be seen in the figure. Prove that the middle quadrilateral area is one ninth of the whole area.
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Find the angles of given triangle ABC

A triangle $ABC$ with angle bisectors $AA_1$ and $BB_1$ is given, such that $\angle AA_1B_1 = 24^\circ$ and $\angle BB_1A_1 = 18^\circ$. Find the angles of the triangle. I've been stuck on this one for quite a long time. After denoting with $I$…
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Given three non-overlapping circles, can we construct (via straightedge and compass) the triangle of minimum perimeter with one vertex on each circle?

G. Polya "Mathematics and plausible reasoning" Chapter 9, problem 2: Three circles in a plane, exterior to each other, are given in position. Find the triangle with minimum perimeter that has one vertex on each circle. From the contents of the…
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Finding tangents to a circle with a straightedge

There is a geometric construction that I heard years ago and I still haven't figured out why it works despite several attempts. Playing with pen, paper and GeoGebra makes me confident that it does indeed work. Could someone explain it to…
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Hole in the axioms of Hartshorne's "Foundations of Projective Geometry"?

I'm currently working my way through Foundations of Projective Geometry by Hartshorne, and he states the axioms characterizing an affine plane as: An affine plane is a set $\mathbb{X}$ together with a collection…
Alec Rhea
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Covering a polygon with an odd number of sides

I have the following elementary problem/question that I do not know how to tackle. It comes with a "math-olympiad-flavor" but I suspect it may be much more difficult than an high-school olympiads problem. Let $\mathscr{P}$ be a simple polygon (not…
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