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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1answer
245 views

What's a good text to read before Coxeter's Geometry Revisited?

I am interested in reading Coxeter's famous text Geometry Revisited. It's not clear to me what the prerequisites for this text are, however. I'm sure I have enough mathematical maturity: I know ...
3
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1answer
78 views

Parallelogram and Areas

Consider the parallelogram $ABCD$. On sides $BC$ and $CD$ take points $E$ and $F $ respectively such that $\frac{BE}{EC} = \frac{CF}{ FD}$. If the segments $AE$ and $AF$ cut $BD$ at $K$ and $L$, show ...
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1answer
245 views

Justify each step in the following proof of Proposition 3.9 (a). I like to know if I'm in the right track and I'm also missing a few of them

Proposition 3.9(a): If a ray r emanating from an exterior point of triangle ABC intersects side AB at a point between A and B, then r also intersects side AC or side BC. proof. (a) Let r= array XD ...
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1answer
79 views

Measure of angle formed by chords and two circles

The following is a question from a practice GRE Math Subject Test: In the Euclidean plane, point A is on a circle centered at point O, and O is on a circle centered at A. The circles intersect at ...
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1answer
46 views

Determine if this interpretation satisfies axiom Congruent axiom 1

Recall the interpretation of the rational plane: points are ordered pairs $(x, y)$ with $x, y \in \mathbb{Q}$; lines are solution sets of equations $ax + by + c = 0$ with $a, b, c \in \mathbb{Q}$ and $...
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2answers
117 views

Prove that OD is a the angle bisector of the angle BOC.

Let $ABC$ be a non-isosceles triangle and $I$ be the intersection of the three internal angle bisectors. Let $D$ be a point of $BC$ such that $ID \perp BC$ and $O$ be a point on $AD$ such that $IO \...
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2answers
43 views

Area of triangle interior to parallelogram

Hi you can help me with this exercise? I have to find the area of the triangle $QOP$ in terms of the parallelogram $ABCDB$ but I do not know how to prove that area triangle $COD$ is $1/8$ of $ABCD$ ...
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2answers
150 views

Get canonical equation of ellipse

We have an ellipse with a circle in it. The circle is passing through the two vertices and through the ellipse's center. It's diameter equals 7. We have also an ...
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1answer
44 views

Do such triangles exist?

Are there two triangles with equal angles and two equal sides which are not congruent? I think it is impossible.
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301 views

the set of points equidistant from $ u $ and $v$ form a line.

Let $u$ and $v$ be two vectors in $ \mathbb{R}^2 $ with the standard norm. Show that the set of points equidistant from $ u $ and $v$ form a line. I show that if $x$ is equidistant from $u$ and $v$, ...
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1answer
62 views

Do two lines lay in the same 4-dimensional plane

I don't know how quite to phrase this, but I'll try. Because two point are co-linear and two lines cannot always be used to define a plane and aren't always in the same plane, are two lines always co-...
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2answers
409 views

How to show that five points in ℝ³ are cospherical?

There are many conditions equivalent to the cocircularity of four points on a plane, however i could not find any such lists for the three-dimensional analog. When do five points in three-dimensional ...
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1answer
471 views

Largest $n$-vertex polyhedron that fits into a unit sphere

In two dimensions, it is not hard to see that the $n$-vertex polygon of maximum area that fits into a unit circle is the regular $n$-gon whose vertices lie on the circle: For any other vertex ...
0
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1answer
21 views

How could we define the existence of an object/element in the Euclidean space?

Let X be an object/element, What does it mean when I say "X is an object in the Euclidean space"? in other words, What differs an existed object from an unexisted one in the Euclidean space?
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1answer
37 views

Euclidean Geometry Construction

I am looking for an aswer to the following construction construct a triangle given two angles (3 angles) and the sum of two sides
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0answers
25 views

Incidence Geometry

Consider a quadraple $(a,b,c,d)$ of points in the real plane such that $|ab| = |cd|$. If the perpendicular bisector of line segment $ac$ is parallel to perpendicular bisector of $bd$, then how does ...
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1answer
48 views

Fundamental domains of Dihedral groups

Let $D_n$ be dihedral group of order $2n$, it acts on plane $\mathbb{R}^2$ in a standard way, by rotations and reflections. How one can find fundamental domains for such action?
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1answer
84 views

Given three concurrent lines $a,b$ and $c$, find the circunference tangent to $a$ and $b$ and with center at $c$

I have these three lines, and I need to construct a circumference tangente to two lines and that has center at the other line. I tried to construct the perpendicular lines that passes through the ...
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2answers
23 views

Given two points $A$ and $B$ and two distances $m$ and $n$, find a point that has distance $m$ fom $A$ and $n$ from $B$

I know that, as long as the distance from $|GI|<m+n$, as you can see in the figure $1$, I can constructo such point by the intersection of the circles with center at $G$ and radius $m$ and with ...
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1answer
35 views

Determine the isometric group $G$ which transfers a square into it self

I am solving the following exercise: Determine the isometric group $G$ of the euclidean plane which transfers a square into it self. The restriction of an element $g \in G$ on the vertices of ...
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1answer
128 views

Euclidean geometry question

Let $(P,L,\varepsilon)$ be a plane with finitely many points (i.e $P$ is finite) Assume in addition to the axioms of incidence that for each $Q \in P$ and $l \in L$ with $Q \not\varepsilon l$ there ...
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1answer
40 views

How many vectors exist satisfying the angle between any two vectors equals to a constant $\beta$ with $0<\beta<\pi$ in a $n$-dimension Euclid space?

At first, if $\beta=\pi/2$, we know that at most $n$ such vectors exist, that is, orthogonal vectors. It's obvious that the number of vectors is influenced by the angle $\beta$. Assume we've already ...
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2answers
210 views

Stupidly simple geometry problem I can't do

Okay. Here it goes. C and D are two points on the same side of a straight line AB and P is any point on AB. Show that PC + PD is least when the angles CPA and DPB are equal. I have no idea why I ...
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1answer
280 views

Locus problem solve it using simple mathematics

A line cuts $X$-axis at $A(7,0)$ and the $Y$- axis at $B(-5,0)$. A variable line $PQ$ is drawn perpendicular to $AB$ cutting $X$-axis at $P$ and $Y$-axis at $Q$. If $AQ$ and $BP$ intersect at $R$, ...
0
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1answer
36 views

What goes the Pappus' theorem says

I found the following statement: Let $A, C$ and $E$ be three distinct points on the line $l_1$ and $B,D,F$ three distinct points on the line $l_2$. Let us assume that $AB\cap DE=L$, $CD\cap FA=M$, ...
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1answer
19 views

Solving the euclidian distance squared to kernelize a Lagrangian dual

Homework question, looking for a hint on the following problem: I'm trying to solve this dual lagranging form (which could potentially be wrong already, but let's assume it is right) $\boldsymbol{x}$ ...
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2answers
532 views

How to find the number of squares formed by given lattice points?

Let us say that we are N integer coordinates (x, y) - what would our approach be if we were supposed to find the number of squares we could make from those given n points? Additionally, if we were to ...
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1answer
97 views

Triangle orthocenter problem

I found a theorem written in a clumsy way. Is this theorem true? Let $ABC$ be a triangle and $DEF$ triangle made by the base points of altitudes of $ABC$. Then the center of an incircle of $DEF$ is ...
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1answer
32 views

Find the line that passes by $P=(1,-2,3)$ and has angle $45$ and $60$ respectively with the $x$ and $y$ axis

I have $$P=(1,-2,3)$$ And the x-axis with direction vector $(1,0,0)$ and y-axis with direction vector $(0,1,0)$. By the angle formula: $$\cos\theta = \frac{|\vec u\cdot\vec v|}{||\vec u||||\vec v||...
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1answer
164 views

How to prove the Pitot's theorem?

I read the following Pitot's theorem: A quadrilateral ABCD is tangential if and only if $AB+CD=AD+BC$, where $AB$ means the length of side $AB$. How can I prove it. I mean, the case $ABCD$ is ...
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2answers
209 views

The relation of angle between two slant faces of a pyramid and the angles between slant vectors

Have any of you seen this theorem of relationship of the angles between two slant faces of a pyramid and the angles between slant vectors, provided that two faces of corresponding to $\phi$ and $\eta$ ...
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2answers
165 views

Finding an angle between the side of a triangle and a segment from a point inside the triangle.

Question given below: ABC is a triangle and D is a point inside ABC such that: $$ m(\widehat{DCB})=m(\widehat{CBD})=18^{\circ}\\ m(\widehat{ACD})=24^{\circ}\\ m(\widehat{DBA})=12^{\circ}\\ m(\...
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1answer
29 views

Proof of a point beween two different points. (Geometry)

I'm struggling with some of the logic writing this proof. This is the question: Prove that if X is in AB (AB is a line segment) with X =/= B, then dist(AX) < dist(AB). Logically this makes ...
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1answer
28 views

Angles in a circle

I have troubles to prove the following: Let $\Gamma$ be a circle with center $O$, $a$ be a tangent to $\Gamma$, $A=a\cap \Gamma$, $D$ a point on $a$ and $B\in \Gamma$ such that $D$ and $B$ lies on ...
0
votes
1answer
26 views

Measure of angles is the same

I found the following theorem in a lecture notes without proof: Let $A, B, C, D, E$ and $F$ be points on the plane such that $\angle ABC$ and $\angle DEF$ are either both acute or they are both obtuse....
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2answers
96 views

What are the ranges of triangle angles?

Lets say, that $\alpha \le \beta \le \gamma$. As shown here, $60 \le \gamma \lt 180$. What are the minimum and maximum values of $\alpha$ and $\beta$? The answer: $$0\lt \alpha \le 60 \\ 0 \lt \...
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1answer
51 views

What is the range of angle in front of longest triangle edge?

What is the minimum and the maximum values of the angle $\gamma$ in front of the longest triangle edge?
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1answer
268 views

How to prove the tangent secant theorem

I was reading the the following theorem: Let $A,B$ be two points on the circumference of a circle. Let $C$ be a point outside the circle. Then $\angle BAC=\frac{1}{2}\widehat{AB}$. Is there some ...
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1answer
50 views

Geometric Construction

"Given in position the points $A,B$ and $P$ and a segment $m$, draw through $P$ a line $r$ in such a way that $A$ and $B$ be in opposite sides of $r$ and that the sum of the distances from $A$ and $B$...
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1answer
110 views

Power of a point problem

There was a Finnish matriculation examination there was the following question: Consider a circle and a point $P$ outside the circle. From the point $P$ draw two lines such that each of the line ...
3
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1answer
142 views

Find the maximum possible area for the triangle

Two vertices of an isosceles triangle are (1,2) and (4,6). The inradius of the triangle is $\frac{3}{2}$. Find the maximum possible area for the triangle. My work, for the two possible structures of ...
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1answer
45 views

slope of a curve in $\mathbb{R}^3$

The surface given by $z = x^2 -y^2$ is cut by the plane given by $y = 3x$, producing a curve in the plane. Find the slope of this curve at the point $(1, 3, -8)$. My answer is: $$f(x, y, z) = x^2 - ...
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1answer
40 views

Line goes through the center of circle [closed]

How can I prove this theorem which looks easy. Circle $C$ has a center $O$. Take four points $A,B,D,E$ on the circumference of $C$ such that $AB$ is parallel to $DE$. Let $F$ and $G$ be midpoints of ...
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2answers
153 views

Prove that $\angle BAC + \angle OAP = 180^\circ$

Prove that if you construct two circle centered at O and P and intersecting at A with tangent lines BA and CA. Prove that $\angle BAC + \angle OAP = 180^\circ$. I'm having trouble just starting the ...
2
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0answers
132 views

Derivation of Steepest Descent Direction used in Line Search Methods

In the numerical optimization text I am reading, the Steepest Descent Direction was derived by considering $$ \min_{||p||_2\leq 1} p^T\nabla f(x_k) $$ This resulted in $$ p_k=-\frac{\nabla f(x_k)}...
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0answers
69 views

Synthetic proof of curvature formula

The radius of curvature of a curve $\gamma:I \to \mathbf{R}$ parametric by arc length is $||\ddot \gamma||^{-1}$. I want to demonstrate this using synthetic geometry. Let $A$, $B$ and $C$ be three ...
4
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4answers
336 views

How to show that these two lines are perpendicular?

Let $\triangle AEE'$ be an isosceles triangle with $\angle EAE'=90^\circ$ such that $AE=AE'$ and such that $A$, $E$ and $E'$ lie on the circle $c_1$. Let $\triangle ADD'$ be an isosceles triangle with ...
3
votes
1answer
29 views

How to find these quantities so as to conform to these conditions?

Suppose $a \in \mathbb{R}^k$, $b \in \mathbb{R}^k$. Then how to find $c \in \mathbb{R}^k$ and $r > 0$ such that the following holds? For any $x \in \mathbb{R}^k$, we have $$|x-a| = 2 |x-b|$$ if ...
2
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1answer
53 views

How to prove this assertion about $\mathbb{R}^k$?

Suppose $k \geq 3$, $x$, $y \in \mathbb{R}^k$, $|x-y| = d > 0$, and $r > 0$. Then how to prove the following assertions? (a) If $2r > d$, then there are infinitely many $z \in \mathbb{R}^k$ ...
2
votes
1answer
39 views

Find the equation of line and finding a point in given example

The outer circle is $x^2+y^2=1$ and the smaller circle is $x^2+(y+1-r)^2=r^2$. The arclength is parameterised anticlockwise with $s=0$ at the bottom as shown. If we know $s_n$ and $s_{n+1}$ can we ...