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

learn more… | top users | synonyms (1)

0
votes
1answer
12 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
0
votes
1answer
35 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 which intersect to form the ...
-4
votes
0answers
12 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
0answers
17 views

Check an algorithm to win hex as first player guaranteed

This question has more to do with the validity of the alogirthm than help per se. I'm unsure if this works with all board setups or just this one, or if it's valid at all. I'm going to start with a ...
1
vote
1answer
27 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: ...
6
votes
0answers
69 views

Self-studying Information Geometry

I was recently exposed to the topic of Information Geometry by a friend of mine, and was looking for a good book to begin self-studying this topic. Any suggestions? Also, what subject matter would ...
0
votes
2answers
38 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
5 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
3answers
25 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?
2
votes
1answer
34 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
1
vote
2answers
40 views

Power of a point proof

I found the question on page 13 of this link. Let $P$ be a point inside a circle such that there exist three chords through $P$ of equal length. Prove that $P$ is the center of the circle. I ...
0
votes
0answers
23 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
18 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
0answers
11 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 ...
4
votes
2answers
2k views

What are the vertices of a regular tetrahedron embeded in a sphere of radius R

Imagine you had a sphere of radius R centered at the origin. What are the coordinates of the vertices of the regular tetrahedron which is circumscribed by the sphere? One of the vertices of the ...
4
votes
4answers
906 views

True or False: The circumradius of a triangle is twice its inradius if and only if the triangle is equilateral.

Let $R$ be the circumradius and $r$ be the inradius. The if part is clear to me. For an equilateral triangle, the circumcentre, the incentre and the centroid are the same point. So, by property of ...
0
votes
0answers
24 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 ...
5
votes
1answer
68 views
+200

Locus of a point on a fixed-length segment whose endpoints slide along orthogonal lines

Suppose we have some segment $AB$ of constant length that slides in such a way that its endpoints are moving along orthogonal lines. Let $P$ be a point in the segment so that $|AP| = a$ and $|PB| = ...
20
votes
1answer
471 views

Does there exist regular planar pentagon?

How to prove or disprove that the boundary of any convex body in $\mathbb{R}^3$ (treated as a surface) includes 5 points which form a regular planar pentagon?
0
votes
1answer
15 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 ...
0
votes
2answers
36 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 ...
20
votes
2answers
279 views

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

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 ...
5
votes
1answer
595 views

Cylinder-ray intersections equation

I found an article involving infinite cylinder-ray intersections, and I don't know how they develop this equation: $$(q - p_a - (v_a, q - p_a)v_a)^2 - r^2 = 0$$ In the end of the first page I quote: ...
4
votes
3answers
116 views

Can a figure inside a circle be seen at right angle from any point on the circle?

A convex, closed figure lies inside a given circle. The figure is seen from every point of the circumference at a right angle (that is, the two rays drawn from the point and supporting the convex ...
0
votes
1answer
26 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
2answers
18 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 ...
1
vote
1answer
16 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 ...
0
votes
0answers
36 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 ...
4
votes
1answer
956 views

Equation of right circular cylinder with radius of the base as 2 units.

Obtain the equation of right circular cylinder with radius of the base as 2 units. Its axis passes through $(1, 2, 3)$ and direction cosines are given as $(2, -3, 6)$ I got ...
-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 ...
1
vote
2answers
27 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 ...
-2
votes
0answers
12 views

Proving that a Sierpinski n-simplex has zero n-volume after infinite iterations [on hold]

I am trying to prove that after infinite iterations, a Sierpinski regular n-simplex will always have zero hyper volume. I have defined the process of creating a Sierpinski n-simplex as bisecting every ...
-1
votes
1answer
15 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.
6
votes
3answers
1k views
+100

Splitting equilateral triangle into 5 equal parts

Is it possible to divide an equilateral triangle into 5 equal (i.e., obtainable from each other by a rigid motion) parts?
26
votes
5answers
2k views

Trying to understand why circle area is not $2 \pi r^2$

I understand the reasoning behind $\pi r^2$ for a circle area however I'd like to know what is wrong with the reasoning below: The area of a square is like a line, the height (one dimension, length) ...
1
vote
1answer
19 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
0answers
20 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
15 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
15 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 ...
1
vote
1answer
13 views

calculate circle cardboard segments

I want to make a cardboard lamp, but i want it to look like half a circle. 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 ...
4
votes
1answer
83 views

IMO Shortlist 1995 G3 by inversion

The incircle of $\triangle ABC$ is tangent to sides $BC$, $CA$, and $AB$ at points $D$, $E$, and $F$, respectively. Point $X$ is chosen inside $\triangle ABC$ so that the incircle of $\triangle XBC$ ...
-1
votes
0answers
24 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 ...
3
votes
4answers
71 views

Find the sum of the areas of regions $X$ and $Y$

Right triangle $ABC$ is inscribed in a circle with $AC = 6$, $BC = 8$ and $AB=10$. $AC$ and $CB$ are semi-circles. Find the sum of the areas of regions $X$ and $Y$. This is not so obvious to me. I ...
0
votes
3answers
26 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
15 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 ...
-1
votes
1answer
23 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 ...
17
votes
0answers
94 views

Painting the plane red and blue: Is it possible for each unit circumference to contain exactly $n$ blue points?

I recently stumbled upon the following problem: Consider the plane: You may color each point either red or blue. Is there a way to color it such that each unit circumference (centred anywhere) ...
0
votes
1answer
21 views

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

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 $AB$?
25
votes
9answers
3k views

Books on classical geometry

I'm curious to whether you guys have any tips on book concerning classical euclidean geometry. I'd like somewhat of an advanced treatment, around the same level as Coxeter's "Geometry revisited". I'd ...
0
votes
1answer
53 views