geometry assuming the parallel postulate of Euclid: in a plane, given a line and a point not on that line, there is exactly one line parallel to the given line through the given point.

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Given a triangle $ABC$, with altitude $AD$ and circumcircle radius $R$, show that $AD = 2R\sin\ B\sin\ C$.

Given a triangle $ABC$, with altitude $AD$ and circumcircle radius $R$, show that $$AD = 2R\sin\ B\sin\ C.$$ I'm a bit stumped as to how the altitude of $ABC$ and the circumcircle radius interact ...
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75 views

Find the shaded area

Find the shaded area Here is the equation that i've made \begin{align*} S&=\pi R^2\\ S_1&=\pi {R_1}^2\left(\frac{24}{360}\right)\\ S_2&=\pi{R_2}^2\left(\frac{24}{360}\right) \end{align*} ...
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2answers
72 views

Geometry problem about angles and triangles

I've been working on this problem for a while. It doesn't seem to hard, but I cannot reach a satisfying solution. The triangle $ABC$ is isosceles with base $\overline{AC}$. A point $O$ is also ...
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1answer
32 views

How do I proof that $\angle ABP =\angle AP'B$ and that $P$, $Q$, $Q'$ and $P'$ are on 1 circle?

Given is a circle with center $M$ and a diameter $AB$. $k$ is the tangent to the circle at point $B$. On the circle there are two points called $P$ and $Q$, such that $P$ and $Q$ are both on the same ...
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How do I prove that $CP > \frac 1 2 (AC+BC-AB)$? [closed]

Given is the triangle $ABC$ with point $P$ on side $AB$. How do I prove that $$CP > \frac 1 2 (AC+BC-AB)?$$
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1answer
77 views

A problem of forming equal angles in plane geometry

C and D are two points on the same side of a straight line AB. Find a point X on AB such that angles CXA and DXB are equal.
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354 views

Formula to find the third point of triangle when two points and all sides are known?

I am writing a program in java. I looking for formula to determine the 3rd point in a triangle if the length of all sides and the coordinates of two points are known.
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2answers
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If $ABCD$ is a cyclic quadrilateral, then $AC\cdot(AB\cdot BC+CD\cdot DA)=BD\cdot (DA\cdot AB+BC\cdot CD)$

If $ABCD$ is a cyclic quadrilateral, then $$ AC\cdot(AB\cdot BC+CD\cdot DA)=BD\cdot (DA\cdot AB+BC\cdot CD) $$ I tried using many approaches, but I could not find a proper solution. Can anyone please ...
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70 views

Quadrilateral problem

Assume a quadrilateral $ABCD$ and $M, N$ points on $AB$ and $CD$ respectively, such as $\frac{AM}{MB}=\frac{CN}{ND}$. Lines $AN$ and $MD$ intersect on $K$ and lines $MC$ and $BN$ intersect on $L$. ...
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0answers
183 views

Area of a equilateral triangle given distances of a point in the triangle from the vertices [closed]

A point $D$ inside an equilateral triangle $PQR$. $D$ is located at a distance of $3$cm, $4$cm and $5$cm respectively from $P$, $Q$ and $R$. What is the area of the triangle $PQR$?
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Given a triangle find the length of BC

Given an ABC triangle , if $AB+AD=4$, find the length of BC.
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1answer
186 views

Pythagoras Theorem - Why a Theory? [closed]

Why is Pythagoras Theorem a Theory but not a Law? I mean we use it many times in School and to build stairs etc. and it has been proven however it is called a Theory.
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0answers
40 views

Triangle inequality for angles

For points $O,A,B,C$ in $\mathbb{R}^{3}$, I was trying to show $\angle AOC \le \angle AOB +\angle BOC$. I could show this when all angles were acute. First, I set $O$ to be the origin and $A,C$ to ...
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3answers
142 views

Moscow Math Olympiad 1973

In every polyhedron there is at least one pair of faces with the same number of sides. Solution: Let $N$ be the greatest number of sides in a face of a given polyhedron. Then the number of ...
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38 views

Area of intersection between square and annulus

The annulus' larger radius is $1$, smaller radius is $r>0$, and center is $(0,0)$. The square's sides are parallel with the axes, the lower left corner's coordinates are $(a,b)$, and the upper ...
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2answers
107 views

How many vectors are needed to define a plane in n dimensions?

How many vectors are needed to define a plane/hyperplane in n-dimensional space? In 3 dimensions, if there are 2 vectors with tails at the origin and the heads in differing locations (and the vectors ...
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0answers
60 views

Ratios in a rhombus

NOTE: I am NOT looking for a full answer,just a hint. Last problem on this question. BdMO 2013 Chittagong: Let $ABCD$ be a rhombus.Let $G$ be a point outside the rhombus such that GE is ...
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3answers
89 views

Proof that a line cuts in half the area of a parallelogram iff it goes through the intersection of the diagonals?

I read a theorem in a book which says that a line bisects a parallelogram iff it goes through the intersection of the diagonals. The edge case of this is of course if the line is one of the diagonal ...
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4answers
104 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 ...
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1answer
34 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 ...
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1answer
58 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 $AI^2$.
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3answers
65 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 ...
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3answers
397 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.
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2answers
58 views

How to get radius at any specific point in ellipse

How to find radius of ellipse at any point $(x_1,y_1)$. We know semi-major axis and semi-minor axis i.e. $a$ & $b$. center of ellipse $(x_0,y_0)$. Somewhere I found. $$ r = \frac{ab}{\sqrt{ ...
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38 views

In the change-of-variables theorem, must $ϕ$ be globally injective?

In the above theorem, doesn't $\phi$ need to be injective too? The inverse function theorem merely implies that $\phi$ is locally injective -- is this sufficient? I ask because Marsden, in his ...
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77 views

Why did Euclid propose in a round-about manner?

For example in Book I. Proposition 2 he shows a line between points B and C. He also shows point A somewhere in the vicinity and shows how one would go about recreating that line starting at point A. ...
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2answers
138 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 ?
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1answer
57 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?
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1answer
91 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 ...
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1answer
89 views

The Puzzle of Locating Points in a Quadrilaterally-Faced Hexahedral Creature

The Disclaimer This is NOT homework... I designed the story. I thought a name MIT would be funny, while something along the line of TULSA or SU will still be decent. I know the algorithm to Q3 and 3D ...
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1answer
162 views

Could Euclid have bisected a line segment without his method of superposition?

In Book I Proposition 10 of the Elements, Euclid performs the bisection (i.e. finding a midpoint) of a line segment. In the course of doing so, he uses Book I Proposition 4, the Side-Angle-Side ...
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97 views

Orthornomal matrices [duplicate]

Is there a more direct reason for the following: If the columns of $n\times n$ square matrix are orthonormal, then its rows are also orthonormal. The standard proof involves showing that left ...
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1answer
45 views

Relation between angles in a 3D space

We have two vectors, $u$ and $v$, both of length 1. Let $\alpha$ denote the angle between them. We also have vector $w$ (perpendicular both to $u$ and $v$) and two other vectors, that are connecting ...
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2answers
54 views

Disjoint rectangles with points at their corners

There is a set of $n$ points in the 2-dimensional plane. All x values and all y values are different. We want to draw the largest set of axis-parallel rectangles such that: All rectangles are ...
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1answer
56 views

Credit Given - Geometricly Modeling Infinity with 3 planes and 9 circles - Ratio of Circles

Refer to the attached diagram sketch to help visualize the equation. I am requesting help with an interesting math problem. Basically, I am diagraming infinity using three planes. These planes ...
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109 views

Tough geometry question

I've been working on this problem for a while but I can't seem to figure it out, so any explanations regarding how to solve it would be appreciated. Here it is: Let $AB$, $CD$, and $EF $be three ...
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1answer
55 views

Equivalence classes of points of R^2

Let $A$ and $B$ be two sets of points in $\Bbb R^2$. We define an equivalence relation on the powerset of $\Bbb R^2$, by saying that $R(A,B)$ iff there is a translation $f$ on $\Bbb R^2$ such that the ...
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1answer
31 views

Foci Concentric Circles

My approach: Using the foci formula $$c=\sqrt{a^2-b^2}$$. By plugging in $a=3$ and $b=2$ I obtain plus and minus $\sqrt{5}$. But there's 2 choices with a root 5 result. How do i know which one is ...
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109 views

Fibonacci Sequence or Golden Ratio?

Using the polar coordinate system, $r$ increases directly with $\theta$. In other words, $r=k\theta$. Which of the following shapes is constructed? A) Fibonacci Sequence B) Golden Ratio C) ...
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43 views

What kind of shape?

To construct this shape, draw a circle. Place the compass on a point on the circle and draw an arc of the same radius as the circle. Now place the compass at the intersection of the arc and the circle ...
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1answer
27 views

Figuring out the side of a triangle

I'm having trouble on this problem I don't know how to set it up. I know XO=2 and OB=6. I'd appreciate any hints.
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47 views

Triangle inscribed insemicircle area-ratio question

My approach: $m<A= 60$ degrees and $m<C=30$. This creats a 30, 60, 90 triangle with ratios $$1:\sqrt3:2$$ After getting the ratio's of the areas, I obtain $$\frac{b*h}{\pi r^2}=\frac{1*\sqrt ...
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190 views

Volume of half-cut cylinder

I'm having trouble obtaining the answer for this practice-test problem. I'm taking the volume of the whole cylinder as if it weren't cut then subtract the portion cut off. But I'm not getting any ...
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2answers
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How to maximize (baking) surface area?

I like eating crust, so I am trying different baking molds to try to get the most crust per dough. More generally, I'm interested in the reverse of this more specific question — how to maximize the ...
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1answer
77 views

euclidean geometry books…

I consider myself poor in plane euclidean geometry. so I need a good geometry book which contains very good theory, and a collection a large number of solved problems, and the end of each part.This ...
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116 views

What is the shape of the region where a special point can exist?

For an equilateral triangle $ABC$ which has edge-length $1$, let $D,E,F$ be a point of contact of the inscribed circle and an edge $BC, CA, AB$ respectively. Then, let us call the region surrounded ...
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185 views

Prove that the quadrilateral formed by connecting the midpoints of a quadrilateral is a parallelogram.

Given a quadrilateral $ABCD$, prove that the quadrilateral formed by its midpoints, $EFGH$, is a parallelogram.
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83 views

Inequality regarding areas of triangles

BdMO Nationals 2013: There is a point O inside ∆ABC. After joining A,O; B,O and C,O extend those line and they will intersect BC, AC and AB at points D, E and F respectively. ...
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1answer
51 views

Unit circle - how to prevent backward rotation

Let's assume we have a unit circle (0, 2$\pi$). Basically I have a point on this circle who is supposed to move only forward. This point is controlled by the user mouse and constantly calculate 25 ...
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1answer
51 views

Length of hypotenuse

Let a circle centered at $O$ have radius $OA=10$. Let OB be perpendicular on OA.Let G and E be points respectively on on OB and OA.Let F be a point on the circumference such that GFEO is a ...