For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, angles.

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12
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1answer
484 views

All are sides of obtuse triangles

What is the maximum number of positive integers among which any three are sides of an obtuse triangle? I can find four, $11,11,16,20$. Is it possible to get five or more? We need $a^2+b^2<c^2$ and ...
0
votes
1answer
55 views

German Translation Help

I'm trying to translate one of Hilbert's papers. I'm 90% done, but I am stuck on one sentence, as my German is very poor and the sentence is involved. I hope someone can give me a rough\quick ...
1
vote
2answers
41 views

The Golden Ratio in a Circle and Equilateral Triangle. Geomertic/Trigonometric Proof?

Geogebra gave me 1.61 for the following Golden Ratio construction shown below. Firstoff, has anyone seen anything similar to this construction? Basically begin with an equilateral triangle. ...
2
votes
1answer
29 views

Distance between incentre and orthocentre.

I want to prove that the distance between incentre and orthocentre is $$\sqrt{2r^2-4R^2\cos A\cos B\cos C} $$here $r$ is inradius and $R$ is circumradius. I considered $\triangle API$ ($P$ is ...
0
votes
0answers
21 views

Problem in solving a question of conics.

Let $A$ and $B$ be fixed points in a plane such that the length of the line segment $AB$ is $d$. Let the point $P$ describe an ellipse by moving on the plane such that the sum of the lengths of the ...
0
votes
2answers
17 views

Unseen Theorem Based on triangle

In $\triangle ABC$, $P$ & $Q$ are the mid points of $AB$ and $AC$, $S$ is the mid point of $PQ$ and $R$ is any point on $BC$, prove that :$8\triangle SQR=\triangle ABC$. When I joined $P,C$ then ...
0
votes
0answers
15 views

How does one get $p=2$ from a condition that there be non-trivial linear transformations of every dimension that to any power are $p$-norm-preserving?

Verifying that (p=2) satisfies $$\forall n\in\mathbb{Z}^+.\exists A\in(\mathbb{R}^{n\times n}\setminus\{I_n\}).\forall k\in\mathbb{R}.\forall ...
1
vote
5answers
30 views

Problem based on area of triangle

In the figure, E,C and F are the mid points of AB, BD and ED respectively. Prove that: $8\triangle CEF=\triangle ABD$ From the given, $ED$ is the median of $\triangle ABD$ So, $\triangle ...
0
votes
2answers
16 views

Determine triangle given one side and sum of other two sides

I'm curious if it is possible to solve completely a triangle given one side and the sum of the other two sides. I'm convinced it's impossible, but wanted to clarify. By solve, I mean find all angles ...
5
votes
2answers
266 views

How do squares of non-right triangles relate?

How do squares of the sides of a triangle, any triangle, relate?
0
votes
0answers
12 views

Is isotropy preserved under uniaxial compression?

First consider a cube which contains an isotropic distribution of point particles. The cube has sides of length 1,1,1 parallel to the (Cartesian, orthogonal) x,y and z axes. Next the cube is ...
0
votes
1answer
20 views

Find the perimeter of the given trapezoid

Find the perimeter of the given trapezoid (The diagram is not drawn to scale) I thought I could use the pythagorean theorem, but I have two unknow sides. What do I do now?? Thank you
-4
votes
0answers
38 views

The Golden Ratio in a Circle, Triangle, and Square: simple geometry/trigonometry construction.

I believe that I have found the golden ratio in the below figure. Geogebra is saying it is very close. Might anyone have a geometric/trigonometric proof? An equilateral triangle ABC is inscribed in a ...
0
votes
4answers
34 views

How to calculate the radius of a circle inside a hexagon?

If I know how big is one side of a hexagon, what's the formula to calculate the radius of a circle inside it?
0
votes
0answers
27 views

What is the difference between ≎ and ≏?

I am having a hard time tracking down the meaning behind the ≎ binary relation. ≏ can be found here: https://en.wikipedia.org/wiki/Equipollence_(geometry) And getting a dump from Latex, it is ...
0
votes
0answers
12 views

Can I always replace a 4D - 3D - 2D projection with a 4D-2D projection?

When visualizing a tesseract, we usually use a 3D projection of it. Then the computer screen projects the 3D structure into a 2D image. Is it always possible to replace these two steps with a single ...
0
votes
0answers
20 views

3D rotation Problem [on hold]

Pic1 Pic2 Can anyone explain to me how to rotate in the first pic and second pic? I understand it's trying to do transformation and i understand about the purpose of it and no problem in calculation ...
0
votes
2answers
43 views

2 circles in an isosceles triangle

I've been given the following school problem: ABC is an isosceles triangle (AB = AC). The radius of the incircle is R and of the other circle (which is tangent to the incircle and to the legs of ...
1
vote
1answer
37 views

Prove any line passes through at least two points

I've started reading Introduction to Algebra by Cameron, and I'm stuck on the first exercise. Q. Prove any line passes through at least two points using the axioms given below. Definitions: ...
0
votes
0answers
28 views

How to find all those points whose distance from $x=(2,0)$ is minimum, using $\|x\|=|x_1| + |x_2|$?

The points must be in the closed ball $\{y : \|y\| \le 2\|x\|\}$. I know $|y_1|+|y_2|$ needs to be $\le 4.$ Other than that, I am confused about how to find all the points that are minimum distance ...
2
votes
1answer
35 views

Finding minimum distance between a circle and curve [on hold]

what is the minimum distance between $x^2+y^2=9$ and $2x^2+10y^2+6xy=1$ in Question there is a circle and a curve and we have to find the least distance between them
0
votes
2answers
47 views

Assuming that the sum of the angles of any triangle is 180, how can I deduce Euclid's 5th postulate?

I already did the reverse, namely, if we assume Euclid's 5th postulate, then the sum of the angles of any triangle is 180 degrees. Now I need to show the converse, but I don't really know how to ...
25
votes
6answers
453 views

How do we know the ratio between circumference and diameter is the same for all circles?

The number $\pi$ is defined as the ratio between the circumeference and diameter of a circle. How do we know the value $\pi$ is correct for every circle? How do we truly know the value is the same ...
0
votes
1answer
18 views

Unseen Theorem Based on parallelogram

In the figure, $🔺 APQ=🔺 ABD$, $AB\parallel DC$ and $AD\parallel BC$ then prove that $RC=AP$ My Attempt While trying to show $\triangle ADP=\triangle RCB$ $$\angle DAP=\angle RCB$$ $$AD=BC$$ I ...
1
vote
1answer
28 views

Prove that the sum of the degrees in the interior angles of a polygon with $n$ sides is $180(n – 2)°$.

I would assume this question involves an inductive hypothesis. Show $n=1$ is true. Assume that if $n$ is replaced by $k$, the sum of the degrees in the interior angles of a polygon with $k$ sides ...
1
vote
1answer
17 views

Logarithmic Spiral- N-gon

In the mice problem, also called the beetle problem, $n$ mice start at the corners of a regular $n$-gon of unit side length, each heading towards its closest neighboring mouse in a counterclockwise ...
1
vote
2answers
21 views

Proving the Secant Angles in the Circle

Ok, I know this is a very easy circle geometry problem, but I want to know that how to prove the theorem of angles in the circle. Like this image here: How can I prove that the angle $X$ is the ...
0
votes
0answers
39 views

Riemann mapping theorem - Personal project

I would like to work on the Riemann mapping theorem this summer. Does anyone could give me some good references linked to this objective. For your information, I now currently finishing a degree in ...
-1
votes
1answer
12 views

Does the location of a geosynchronous satellite between two locations on earth matter for total distance traveled of a signal? [on hold]

It's basically you have two concentric circles, with a point on the larger circle called A connecting to two points on the smaller circle, B and C, with distances AB and AC. Does AC+AB change as you ...
2
votes
3answers
31 views

Division of segments into infinitely many parts.

Let AB and CD be two segments, so that the length of AB is 1, and the length of CD is 2. If we divide AB and CD in infinitely many parts, how "long" would those parts be? I'm particularly interested ...
-1
votes
1answer
16 views

how to prove that a figure is a trapezoid? [on hold]

ABCD is a rhombus. let M be any point on [AB] and N any point on [BC]. the parallel through M to BD cuts AD at Q. the parallel through N to BD cuts CD at P. show that MNPQ is an isosceles trapezoid.
1
vote
1answer
12 views

Mirror image of a point about a line

How can I calculate the mirrored position for a point in 2D space? I know the xy-coordinates of the two points which define the vector. I also know the coordinates for the yellow point which I want to ...
0
votes
0answers
22 views

How to use slopes (3 points are given) to prove that they form a right triangle?

Question: Use slopes to show that $A(-3, -1)$, $B(3, 3)$ and $C(-9, 8)$ are vertices of a right triangle. My try at the problem: I know that we can find the slopes of $AB$, $BC$ and $CA$ and then ...
0
votes
0answers
23 views

Unseen Theorem Based on Quadrilateral

In the figure, ABCD is a square and PEFG is a rectangle. If PD=PE, prove that square ABCD and the rectangle PEFG are equal in area. Here $\triangle PCG$ similar to $\triangle PCD$. ...
0
votes
2answers
16 views

If $au + bv + cw = 0$ with $a+b + c = 0$ then $u,v,w$ are collinear

If $u,v,w \in \mathbb R^3$ such that for some $a,b,c$ real numbers with $a+b+c = 0$ we have $au + bv + cw = 0$, then why are $u,v,w$ collinear points? i substituted $a = -b-c$ and tried other things ...
0
votes
1answer
48 views

Find the intersection of plane and sphere

If the equation of the sphere is $x^2+y^2+z^2=1$ and the plane is $x+y+z=1$, then how can the equation of a circle be determined from the equations of a sphere and a plane? and what is the parametric ...
2
votes
1answer
15 views

calculate circle cardboard segments

I want to make a cardboard lamp, but i want it to look like half a sphere. Given a cardboard thickness of x, and a circle width of y, how many elements do I need and what radius do the elements need ...
-1
votes
0answers
29 views

non-transcendental ratio of circumference to diameter

Does there exist a constant non-transcendental curvature of the plane such that the ratio of the diameter to the circumference of a unit circle in that plane is also non-transcendental? Or, if not a ...
1
vote
1answer
22 views

What is the difference between a line segment, and a directed line segment?

Is a line segment by definition directed? Does directed mean it is in movement? If its a segment of a line, doesn't it neccessarily follow the rest of the line?
0
votes
3answers
28 views

How to find the sides of an equilateral triangle given all angles.

How do I find the length of sides and the height of an equilateral triangle when I only know the three angles and the area. The area is 50.3144 and obviously all the angles are 60 degrees. I'm in ...
0
votes
0answers
19 views

Using integration to find volume of a parabolic prism.

Suppose I have a solid with its profile being a parabola given by the equation $y = f(x)$. Its depth at any point is given by $g(x)$ and its width is a constant $k$. This essentially results in a ...
0
votes
1answer
21 views

$Y$ coordinate of a point that lies on a line [closed]

Given two points $A$ and $B$, for example $A(1,5),\,B(15,2)$, what is the $y$ coordinate of a point $C(x,y)$ lying on the straight line $AB$?
-1
votes
1answer
24 views

Question on circles…

If three circles with radii ${3}$,${4}$,${5}$ touch each other externally at points P,Q and R,then the CIRCUMRADIUS of ∆PQR is...?? My attempt i think that the let the point of the common ...
-2
votes
0answers
23 views

To find/create midpoint, is it easier to bisect a line segment, or double a line segment? With only compass and ruler / straightedge. [closed]

Suppose one wants to find the midpoint of a line segment. Is it generally easier to simply draw two lines of equal length end to end, or is it easier (does it count as less steps) to draw a line and ...
0
votes
0answers
18 views

Examples of applications of the Theorems of Pappus and Ménélaüs.

I'm going to present an exposition about applications of the Theorems of Pappus and Ménélaüs. I need some simple examples of these two theorems. Any links, please?
2
votes
1answer
25 views

What is the minimum radius $r$ of two intersecting circles that are spaced $x$ apart that completely enclose a square of length $w$?

Let's say we have two circles whose centers are spaced a fixed $x$ units apart from one another. Both circles have a radius $r$. Our goal is to identify the minimum value of $r$ so that the ...
1
vote
1answer
36 views

Simple Golden Ratio Construction with Three Lines, and Interesting Implied Circle?

Consider three equal lines (as illustrated below). A red, green, and orange line of equal length all rest upon the same horizontal line. The red line is stood upon its end in a manner perpendicular ...
2
votes
1answer
134 views

Which numbers are necessary?

The Greeks were initially convinced that all numbers were rational until upon pain of contradiction were forced to accept that $\sqrt{2}$ was irrational and needed to be included in our number system ...
0
votes
2answers
36 views

Among complex $z$ such that $|z-25i|\leq 15$, which have…

Among the complex numbers $z$ which satisfies $|z-25i|\leq 15$, find the complex number $z$ having: (A) Least positive argument (B) Greatest positive argument (C) Least modulus (D) Greatest ...
10
votes
2answers
197 views
+50

The three unsolved problems of antiquity

In Sidelights on the Cardan-Tartaglia Controversy (Apr., 1938) by Martin A. Nordgaard in the National Mathematics Magazine, Vol. 12, No. 7, pp. 327-364, it is written on the first page The ...