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
vote
0answers
71 views

Classification of Projective transformation according to Jordan form

Say we have the projective space $\mathbb P^2_{\mathbb R}$ = $\mathbb P(\mathbb R^3) \stackrel {\text{def.}}{=} \{\text{span(u)}\mid u\in\mathbb R^3\smallsetminus\{0\}\}$. Denote $[u]$ for an element ...
0
votes
2answers
101 views

How to prove this result about perpendicular bisector?

I have been working on this question for many hours. Thanks for your help. Given: $\overline{BD}$ is the perpendicular bisector of $\overline{AC}$ Prove: $\angle BAC=\angle BCA$ (equal and ...
0
votes
1answer
61 views

Direction of traslation of affine movement

I have a doubt about this. We have an affine isometry of an affine space $X$ of dimension 3. Now, we know it's the composition of some movement (reflection, rotation, etc), with a traslation, and we ...
0
votes
2answers
132 views

Cartesian equation of $ \vec{r}(t)=\left(e^{\omega t}(\cos(\omega t)+\sin(\omega t)),2\omega e^{\omega t} \cos(\omega t)\right) $

I have this parametric equation: $$ \vec{r}(t)=\left(e^{\omega t}(\cos(\omega t)+\sin(\omega t)),2\omega e^{\omega t} \cos(\omega t)\right) $$ and I have to obtain the Cartesian equation. Any ...
0
votes
0answers
35 views

Finding out the value of an angle [duplicate]

Consider the following construction. My goal is to find the value of the red angle. Trying to compute all the angles, starting from the bottom, led me nowhere...
1
vote
1answer
191 views

Completing a very difficult triangle

I have an isosceles triangle with the two equal sides of length 'c', and the bottom of length 'a'. Both base angles of the triangle have measures of 'a', in degrees. For example, if 'a' were 50, both ...
0
votes
4answers
292 views

Minimum distance between a disk in 3d space and a point above the disk

How can I calculate the minimum distance between a point on the perimeter of a disk in 3d space and a point above the disk? For example, there is a disk in 3d space with center [0,0,0]. It has radius ...
0
votes
2answers
198 views

random circle with radius r on cartesian plane, probability of it not cutting x and y axis with intercepts.

I have a tough question here. Choose a circular disk of radius r on the cartesian plane. What's the probability it is not cut by horizontal lines with integer y intercept, or vertical lines with ...
0
votes
2answers
467 views

Question on surface area and volume of cuboid

I came across a question: The surface area of the six faces of a rectangular solid are 4, 4, 8, 8, 18 and 18 square cms. The volume of the solid, in cubic centimetres is __. I can guess that 4 ...
3
votes
1answer
73 views

Is there anything interesting about this figure constructed from a set of points and their barycentre?

Playing with the TikZ package for (La)TeX, I made a nice figure. Well, I think it is nice, anyway. You can ignore the distracting colours and the concentric circles, they are not important for this ...
0
votes
4answers
113 views

How prove this $FC\bot AO$

let $GH$ is the diameter of the circle $O$,and such $$AB\bot GH,FD\bot GH,AB=CD$$ show that $$FC\bot AO$$ My try: since $$AB//FD$$ we only prove $$\angle AOB=\angle CFD$$ or $$\angle BAO+\angle ...
2
votes
1answer
477 views

What is the value of $\angle x$ [duplicate]

Can any one help with this problem:what is value of $\angle x$
0
votes
1answer
120 views

Finding the point of where three fixed-length lines intersect

We know the values of the coordinates (Xa,Ya), (Xb,Yb), and (Xc,Yc). We also know the lengths of A, B, and C. Is there a way (equation) to figure out the exact coordinates where the three lines A, B, ...
2
votes
3answers
4k views

construct circle tangent to two given circles and a straight line

How to construct such a circle? Only straight edge and compasses are allowed.How could we draw if we were to draw the circle tangent to two straight lines and a circle.
0
votes
1answer
61 views

preimage of a semi-circle under $p(z)=z^2$

Let $e^{i\theta} \in S^1$, consider $f:\Bbb R \to S^1$ given by $f(t)=e^{it}$. Let $U$ be the image of the set $(\theta-\frac{\pi}{2},\theta+\frac{\pi}{2})$ under $f$ it's the open semicircle centered ...
1
vote
2answers
62 views

Relation between the radius and the area of tangential polygon

I've recently found a book with loads of formulas for triangle area, but unfortunaly the formulas were just listed, there wasn't a proof for them. I've tried to proof them. But I've stopped at one of ...
0
votes
1answer
94 views

Proof adding layers of constant width to a shape tends to an $d$-sphere as the number of layers tends to $\infty$

Good night, I've recently seen one of Victoria Hart's videos on Youtube (it wasn't about this, it was about Fibonacci numbers, and I found it on a comment in this site), and in it she said that if ...
0
votes
1answer
159 views

what is the most sided sturdy regular n-polygon that can be made with lego?

Was puzzeling with this question: What is the most sided regular n-polygon that can be made with lego? It has to be sturdy (the polygon should stay in shape when pushed around) made with the normal ...
0
votes
5answers
736 views

Mensuration question about a hoop resting on a staircase

I came across a question recently: A hoop, as shown in the diagram, rests vertically at stair case. Note: AB = 12 cm, and BC = 8 cm. Find the radius of the hoop. Figure (hand-made): This is ...
1
vote
5answers
483 views

Mensuration question

I recently came across a puzzling question: Two rectangles ABCD and DBEF are as shown in the figure. The area of DBEF is: Figure (hand-made): I know that through Pythagoras, we get ...
0
votes
1answer
39 views

Geometry question with convexity

Assume that a function $h(\lambda)$ is decreasing and convex given interval $[l,u]$ and has an unique root $\lambda^*\in (l,u)$. Also, assume $|l-\lambda^*| > |\lambda^*-u|$. Consider any $z\in ...
0
votes
2answers
118 views

What is a Solid Angle?

What is a Solid angle.How do we measure a solid angle? How is it different from a plane angle and how do we construct a solid angle
1
vote
1answer
84 views

Angle bisector in a triangle

For the angle bisector $I_a$ in a triangle $ABC$ it holds $$I_a^2 = \frac{bc}{(b+c)^2}[(b+c)^2 - a^2]$$ If $I$ is the incenter, I wonder if there exist similar formula for the part $AI^2$.
1
vote
1answer
192 views

Maximal size of triangulation in 17-gon

Given convex 17-gon. What is the maximal count of triangles we can divide it if we draw all it's diagonals? (for 4-gon,answer is 4, for 5-gon answer is 11)
3
votes
3answers
77 views

Simple proof that symmetries of regular polyhedron fix its center?

Let $P$ be some regular polyhedron in $\mathbb{R}^3$ (i.e. a regular $n$-hedron with $n = 4, 6, 8, 12,$ or $20$), centered at the origin $o = (0, 0, 0)$, and with vertices $v_1, ..., v_n$ all lying on ...
2
votes
2answers
550 views

Ellipse bounding rectangle

I'm trying to find the ellipse that bounds a rectangle in a way that the "distance" between the rectangle and the ellipse is the same vertically and horizontally. Here is an image to illustrate what ...
3
votes
3answers
2k views

Book with lots of geometry theorems

I want to study geometry and was looking for some book that has lots of theorems and covers almost all Euclidean geometry that is needed for High School and Maths Olympiads. Thanks.
0
votes
1answer
169 views

Geometry question on Trapezoid.

The lengths of the bases of a trapezoid are represented by $x-10$ and $3x-8$. Express the length of the median of the trapezoid in terms of $x$.
1
vote
1answer
528 views

finding centroid of polygon with holes (polygons)

I am able find the centroid of polygon without holes. How can I want to find centroid of polygon with holes(polygons itself). will the centroid will come outside holes or inside. please help me on ...
0
votes
2answers
181 views

How do you find the intersection point between a ray with a $2\text{D}$ line?

If I have a ray that has a $\text{position}(x, y)$ and a $\text{direction}(z, w)$ and a line that has a $\text{start}(j, k)$ and an $\text{end}(u, v),$ how do I find the intersection between the two?
5
votes
1answer
831 views

How is the Radian measure of angles derived/defined?

I'm currently studying the foundation of trigonometry (angles and their measures) and I've just been told that $\pi$ is the ratio of a circle's circumference to its diameter, so: $\pi =\dfrac ...
2
votes
2answers
64 views

Curvature of a plane given a circumference

I was skimming through a certain book when I came to an interesting passage. And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five ...
0
votes
3answers
46 views

Determining rectagle width and height, when given perimeter and space

What would be the formula, to determine a rectagles edges, when given the perimeter and space? for example, the rectagles space is 80, and the perimeter is 36, and the edge would be 8 and 10, but how ...
0
votes
1answer
137 views

Proving properties of an ellipse

I'm studying about ellipse and its properties. My reference is the following pdf: http://nebula.deanza.edu/~bloom/math43/ellipse-derivation.pdf My questions are from the very first page of the ...
1
vote
0answers
55 views

Help understand this proof: unit disk on plane have no finite paradoxical decomposition

I am working through the note here. I have been stuck at Proposition #5 proof for a while because I can't understand the last part. Specifically, the proposition said that "The unit disk $D$ in ${\Bbb ...
2
votes
0answers
81 views

Geometric construction of catenary

Can anyone explain the steps of the Leibniz geometric construction for the catenary curve? Leibniz does a complete job, I'm sure, but I still cannot follow with certainty; see ...
2
votes
2answers
304 views

Find value of the angle x

Find the value of the angle x. Plus : Someone could recommend me some good book about this subject ?
1
vote
1answer
348 views

Find the function that describes a real life curve

Let's say you want to compute the length of an arbitrary 2 dimensional curve or the area between two arbitrary 2 dimensional curves in the real world. For example you take your pen and draw a line and ...
3
votes
1answer
66 views

Formula in a triangle

Let $H$ be the orthocenter in a triangle with sides $a, b, c$. Is it true that $$a^2 + HA^2 = 4R^2$$ where $R$ is the circumradius?
0
votes
2answers
719 views

Distance between centroid and incenter in a right-angled isosceles triangle

Let ABC be a right-angled isosceles triangle where AB = BC = a. Assume that C is its centroid and I is its incenter. Find, in terms of a, the distance between C and I. Answer : $CI= \frac{{a \cdot ...
1
vote
1answer
537 views

Calculating angle of rotation of orthogonal 3x3 matrix

Regarding the matrix in Q3b here: http://www.maths.ox.ac.uk/system/files/coursematerial/2013/2637/5/13sh2.pdf I've worked out the axis of rotation by finding out the line of invariant points, but I'm ...
5
votes
1answer
170 views

Rational distance from an equilateral triangle

Is there a nice proof for the following fact? In a plane, there does not exist a square such that its vertices are at a rational distance from each vertex of some equilateral triangle. What if ...
0
votes
2answers
1k views

Software to draw easily sectors with angle on it

I want to draw a sector with the angle on it. I have tried several tools but didn't find any easy way of doing it.
1
vote
1answer
623 views

How to find vector equation of a plane given a line and a point?

I have the following information: a.b = 0 where . represents the scalar product. The plane contains the line r x a = b where r is a general point on the line and x represents the cross product ...
2
votes
1answer
312 views

The formula for pitch circle diameter.

I want to put $n$ number of circle with $r$ radius each in a big circle. Want to calculate the radius $R$ of the big circle. How can this be achieved?
7
votes
3answers
669 views

Understanding cross ratio and harmonic conjugates

I'm studying projective geometry and I'm really having trouble with ''grokking'' what's it all about. Is there an easy/intuitive/visual way to understand cross ratio? I understand that it's ...
1
vote
4answers
1k views

Quarter-Circle inscribed in a Square.

Quadrilateral ABCD is a square with a side length of 4 cm. A quarter-circle with radius 4 cm and centered at D connects A and C, and a quarter-circle with the same radius centered at B also connects A ...
0
votes
1answer
84 views

Splitting a Rectangle into 2 halves and getting the dimenions of the top halve.

Hey im making a game and i need to get the dimensions of the top half of a rectangle. Say i had a rectangle with the height of 15 and the width of 10. Firstly how would i split it into 2 sections, ...
-1
votes
1answer
97 views

find a common plane which contains two points (NP hard?)

In this problem the coordinates of 4 points are given. $p_{0}=(x_{0},y_{0},z_{0})$, $p_{1}=(x_{1},y_{1},z_{1})$, $p_{2}=(x_{2},y_{2},z_{2})$ and $p_{3}=(x_{3},y_{3},z_{3})$ I need to find the ...
2
votes
1answer
103 views

Inequality in a triangle

Let $O$ be the circumcenter and $H$ the orthocenter in a triangle with sides $a, b, c$. Is it true that $$aOA^2+bOB^2+cOC^2 \ge aHA^2 + bHB^2 + cHC^2$$ or equivalently $$(a+b+c)R^2 \ge aHA^2 + bHB^2 + ...