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28.550 geometry questions.

This is a question I met in some notes. I need some help here. Let $ABC$ be an acute triangle and let $AA_1, BB_1, CC_1$ be its altitudes. Segments $AA_1$ and $B_1C_1$ meet at $K$. The ...

I need to find the area of the shaded (grey) region in the above picture. All sides of the cross are 4. I feel like this question has an easy answer, but I cannot seem to figure it out. I tried to ...

I remember seeing somewhere that for every connected compact set $\Omega$ in $\mathbb{R}^2$ with piecewise $C^1$ boundary we have $$A(\Omega_r)\leq A(\Omega)+L(\partial \Omega)r+ \pi r^2,$$ where $$\...

Given the unit square, it is clearly possible to dissect it into any even number of triangles of the same perimeter (the triangles being even isometric). But I wasn't able to find such a dissection ...

We are given angles A and B (70 and 60 respectively). Also AΓ=ΒΔ. Ζ and E are midpoints of AB and ΓΔ respectively. I also drew some bigger circles with radius AH and ΒΘ, trying to see some pattern ...

In the Euclidean plane, one can define the so-called root-2 rectangle, i.e. a rectangle whose side lengths are in the ratio $1 : \sqrt{2}$, or 1 : 1.414 (3.d.p.); a key property is of being divisible, ...

An ant wants to get from (0,-2) to (0,2). However, there is a cone of radius 1 and height h at (0,0). What is the length of the shortest route? Going around the cone uses $\frac{\pi}{3}+2\cdot \sqrt{...

square1 = (47,15) (51,15) (51,19) (47,19) square2 = (19,0) (-27,46) (19,92) (65,46) These two squares may not be parallel to X axis or Y axis. These two squares do not intersect, but how can I find ...

We have an isosceles triangle and we divide it by two sections going out of one of three corners, hence we get three new triangles. Is it possible to make (puzzle) an isosceles triangle out of every ...

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In the triangle $ABC$, $\angle A = 90$°, the bisector of $\angle B$ meets the altitude $AD$ at the point $E$ and the bisector of $\angle CAD$ meets the side $CD$ at $F$. The line through $F$ ...

I have a plane $P$ that intersects a sphere $S$ with a radius $R$. The result of this intersection is a circle $C$ with the distance between its center and the sphere's center is $s$ In my case is ...

If given 3 points such as O(0,0) B(3,0) A(4,3) how to find bisector OAB angle line equation?

Find the area of the parallelogram whose perimeter is $20$, one internal angle is $30^\circ$ and heights are in the proportion $2:3$. Setup So, the perimeter is $20$: $$P=20$$ One angle is $30^\circ:...

Find the area and the angles formed by the diagonals of the rhombus, if its height is two times smaller than the length of its side and is $4.2$. My attempt Let's write down the height: $$h=\frac{a}{...

The Erdos-Szekeres theorem (happy ending problem) says that among any $n$ points in general position in $\mathbb{R}^2$, there are $\Omega(\log n)$ points in convex position. There are also ...

Let $A,B,C,D \in \mathbb{RP}^2$ be a quadrivertex. Let $f: \mathbb{RP}^2 \to \mathbb{RP}^2, A\mapsto B, B \mapsto C, C \mapsto D, D \mapsto A$. Find the fix points and invariant planes of $f$ and ...

Two identical right circular cones are resting on their curved surface as shown in the second row of the image above. Let V be the top vertex and O (O') be the center of base circles and A(A') be the ...

How did the ancient Greeks discover formulas for volume and surface area of different objects, e.g. of a sphere? They did not know about integrals, so there must another way?

this is the first time I ask a question here. I want to know what's the simplest way of calculating the coordinates for the 4 points of a square/rectangle on a plane, after a rotation around a point, ...

Prove that the perpendicular bisectors of the interior angle bisectors of any triangle meet the sides opposite the angles being bisected in three collinear points. Here is what I have so far. ...

Find the length of AC What i tried Since the diagram is a trapezoid, angle $DAB$ is $180-140=40$ degrees. Then using the properties of simillar triangles, angle $DAC$ is equal to angle $CAB$,thus ...

A measure $V$ (for "value") is defined on an equilateral triangle. $V$ is absolutely-continuous with respect to the Lebesgue measure, and the value of the entire triangle is $1$. What is the largest $...

I recently watched a video on a youtube channel "DONG" led by Michael from Vsauce titled "Making every strictly convex deltahedron" Some concepts like elongation, gyroelongation, expansion, ...

"Be a circumference of radius $R$ and center $O$. A second circumference is drawed in such a way that is tangent to the first circumference (in a point $P$) and goes through the center of the first ...

A rectangle is 4 times as long as it is wide. If the length is increased by 4 inches and the width is decreased by 1 inch, the area will be 60 square inches. What were the dimensions of the original ...

I have two coordinate frames moving with respect to each other. One frame is attached to a camera such that the $z$-vector is in the direction of the axis of the camera. The other frame is attached to ...

In the diagram, $CE=CF=EF,\ EA=BF=2AB$, and $PA=QB=PC=QC=PD=QD=1$, Determine $BD$. I tried working this question as : $$\angle ACB = \angle DQB = 12^\circ$$ So, by the cosine rule for $\triangle ...

Given: the situation described in the figure below, with $\alpha=45^\circ$, $\overline{AS}=12$, $\overline{DS}=6$. Find: $\overline{QS}$. Question from a math contest. I've tried some ideas but ...

The answer should be I) and II), but I have only been able to solve II). It is supposed that I must solve this only with knowledge of secondary elements of the triangle and criteria of congruence of ...

The segments can also cut each other. I have tried and tried but I can do it with 5 segments only.

I am studying affine and projective geometry and I have encountered some invariant: the cross ratio, which in Italia is called "birapporto" and another one which I do not know the name of in English. ...

Next year I will be embarking on a dissertation on isoperimetric inequalities; I'm doing some initial research and I am frequently coming across the definition; Let $\Omega$ be an open set in $R^n$. ...

I cannot conceive of any possible way this could occur, dDo any of these kind of lines exist?

In the real affine space $\Bbb A^4$ consider the plane $\pi$ defined by $$\begin{cases}x-y+w-1=0 \\ x-2y+z+1=0 \end{cases}$$and the line $r_k$ that goes through $P=(1,0,3,1)$ and $Q_k=(2k,k,3k,2k)$, ...

Prove that the triangle inscribed inside a circle has the largest area geometrically.(Without calculus). How can i explain this to a class 6 child.

I read on the Internet it's true, but I suspect it: Take a ribbon tightly wound around the equator of the earth. Add 1 meter to that ribbon by cutting it at any point. Uniformly lift above the earth, ...

Probably really dumb question but here it goes. We know Volume of solid formed by rotating $f(x)$ along x-axis is given by $$V=\pi\int_a^b(f(x))^2dx$$ Which is derived by adding surface area of ...

How can we find the area of the rhombus with side length $a=5$ and sum of diagonals $d_1+d_2=14$. I know two formulas for the area of rhombus ($h$ stands for height): $$A=ah=\frac{d_1d_2}{2}$$ But I ...

Hi, I'm struggling to complete this question. I am required to find vectors $\vec{ED}$ and $\vec{EF}$ in terms of a and b. I have so far found that vector $\vec{ED}$ = b - $\frac{5}{3}$a. I would ...

We can see that shaded region is area of FEH minus the sector HGE. To find the sector HGE i called the angle GHE as $\theta$ and used $$ \pi\times\left(\frac{\theta}{360^{\circ}}\right) $$ If FE is $...

Why is it considered that in x-y plane counterclockwise movements are positive and clockwise are negative? And in x-z plane it is vice versa? How can I know which movements are positive and negative ...

How can I find the ratio of the volume of a region in the unit n-dimensional hypersphere where for a radius $0<r<1$, the region is bounded such that $r^2<\sum_{i=\{1...n\}}x_i^2<1$ ; $\...

Find the equation to the common conjugate diameters of the conics $x^2+4xy+6y^2=1$ and $6x^2+6xy+9y^2=1$. I dont know how to start here.

For a coordinate system $C_2$, if I have the angles made by its coordinate axes ($x_2$,$y_2$ and $z_2$) with the coordinate axes($x_1$, $y_1$ and $z_1$) respectively of another coordinate system $C_1$,...

The points $P$ and $Q$ are chosen on the side $BC$ of an $acute$$angled$- $triangleABC$ so that $\angle PAB=\angle ACB$ and $\angle QAC=\angle CBA$. The points $M$ and $N$ are taken on the rays $AP$ ...

I recently saw a similar problem online but found the area in a completely different way. Problem: There is a unit square, i.e. a square with side length equal to $\text 1$. Two lines are placed ...

A circle is divided into 360 little parts called degrees. Why or how did they choose that figure? Is there a very strong reason for that or it just a accidentally choose.

This might be an odd question, but I recently spent a very long time solving a problem in Gamelin's Complex Analysis and I wanted to understand exactly what was making it so difficult for me. ...

I am trying to get a best fit plane in a 3d space of points. I am using an svd as described in https://stackoverflow.com/questions/10900141/fast-plane-fitting-to-many-points. If I use the data ...

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