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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Prove that $CDEF$ is a rectangle

Two circle $\Sigma_1$ and $\Sigma_2$ having centres $C_1$ and $C_2$ intersect at $A$ and $B$. Let $P$ be a point on the segment $AB$ and let $AP\ne{}BP$. The line through $P$ perpendicular to $C_1P$ ...
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1answer
55 views

Prove that the circumcenter of a triangle lies on an angle bisector

Let $\triangle$ ABC be a triangle and let $\ell$ be the A-angle bisector. Denote by B' the reflection of B over $\ell$. Prove that the circumcenter of $\triangle$ CIB' lies on $\ell$. My work: Let D ...
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1answer
25 views

Parabola with given points

Let if there is a parabola passing through some points eg $(0,1)$ , $(-1,3)$ , $(3,3)$ & $(2,1)$ Then if have we to find vertex and directrix . As there are two parrallel chords then the abcissa ...
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1answer
27 views

How to write sum notation for an array of 2D points

What is a correct way to write using sigma notation for a problem involving an array of 2-dimensional points. Say I have 2 arrays, $P_{e}$ and $P_{a}$, both containing $N$ elements. $P_{e}$ represents ...
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1answer
25 views

Focal distance on a parabola

If focal distance of a point on the parabola $y = x^2 - 4$ is $25/4$ . And the points are of form $(\pm \sqrt{a}, b$ then how can we find $a , b$ or sum of these . I think the focus of parabola would ...
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51 views

Problem with Cauchy Schwarz equality proof

Can anyone help me with this proof? I'm a bit confused about it. I want to show that in a finite dimensional euclidean vector space $(V, \cdot)$ for two vectors $\vec{x}$ and $\vec{y}$ is true that ...
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3answers
88 views

What is the angle $\widehat{BAC}$?

Let the triangle $ABC$ and the angle $\widehat{ BAC}<90^\circ$ Let the perpendicular to $AB$ passing by the point $C$ and the perpendicular to $AC$ passing by $B$ intersect the circumscribed ...
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2answers
28 views

Proving area of a square inside of two squares in Euclidean Geometry

Let ABCD and PQRS be squares of the same side length such that P is the center of ABCD (i.e., the intersection of its diagonals). Suppose that BC and PQ intersect each other at X. Suppose also that ...
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65 views

A triangle perspective with many well-known triangles [closed]

Let $ABC$ be a triangle, construct a circle $(O_a)$ touching to $B$-excircle, $C$-excircleand inside the circumcircle at $A_b, A_c, A$ repectively. Define $B_c, B_a$ and $C_a, C_b$ cyclically. Let ...
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How to find all glide-reflections-symmetries in a regular point-lattice.

Let $A,B,C$ be non-collinar points in the euclidian plane $\mathbb{E}$ and $\tau_{A,B},\tau_{A,C}\in Iso(\mathbb{E})$ translations moving $A$ to $B$ alternatively $A$ to $C$. So... ...
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62 views

Find the length of AB

In the diagram 4 circles of equal radius stand in a row in such a way that each circle touches the next one. $P$ is a point on the circumference of the first circle. The center of the fourth circle is ...
4
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1answer
103 views

Three problems of ten circles

Problem $1:$ Let four (red) cirles $1, 2, 3, 4$ such that $1$ touching $2, 2$ touching $3, 3$ touching $4, 4$ touching $1$ and $1, 2, 3, 4$ touching (blue) circle $5.$ Construct four purple circles ...
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72 views

The reverse pizza problem .

The pizza problem is a fairly well-known problem which sounds like this : You have a circular pizza and you need to cut it such that you and your friend would both receive half of the pizza . ...
4
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1answer
80 views

Go from A to D in three equal steps

Given two parallel lines $r$ and $s$, line $p$, perpendicular to both, and points $A$ and $D$ on different sides of $p$ with respect to the parallel lines, how can I prove the existence of two points, ...
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1answer
36 views

Parabola conic section

Two tangents to the parabola $y^2= 8x$ meet the tangent at its vertex in the points $P$ and $Q$. If $|PQ| = 4$, prove that the locus of the point of the intersection of the two tangents is $y^2 = 8 ...
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1answer
24 views

Finding vertices of a rhombus in complex plane

If $A,B,C,D$ are the vertices of a rhombus and $M$ is the point of intersection of the diagonals such that $2AC=BD$, find the vertex $A$ if $D$ is $1+i$ and $M$ is $2-i$. Attempt: $$MA\perp MD$$ ...
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22 views

Geometry: Inequalities are preserved under addition of line segments.

This is a question from Hartshorne's Geometry: Euclid and Beyond. For this problem you are supposed to use the usual congruence axioms and notions of betweenness as well as the incidence axioms. ...
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7 views

Existence of the Schiffler Point

Let triangle $\triangle{ABC}$ have incenter $I$. Prove that the Euler lines of $\triangle{AIB}, \triangle{BIC}, \triangle{CIA}, \triangle{ABC}$ are concurrent.
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75 views

Proof: Angle between two diagonals of a parallelogram

Can anyone prove the angle $\theta$ (smaller angle) between diagonals of a parallelogram is given by the equation $$\cos(\theta)=\frac{a-c}{a+c}$$ where $2a$ is the base length of parallelogram; ...
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1answer
44 views

circles, power of point, cross ratios

Let $w$ be a circle, and let $P$ be a point outside $w$. Let $X, Y$ be the tangents from $P$ to $w$. A line from $P$ intersects $w$ in two points $B, D$. Let $C$ be the intesection of $\overline{XY}$ ...
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35 views

Composition of a rotation and translation is a rotation with what angle?

Im trying to figure out what degree exactly you get when you first translate and then rotate or the other way around. I'm trying it without coordinates in the Euclidian plane. Let $\delta\in ...
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1answer
139 views

Prove that every triangle is the orthogonal projection of an equilateral one

Prove that every triangle is the orthogonal projection of some equilateral triangle. This problem appears in a book I'm working through in the chapter on transformations in space. There is a rather ...
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1answer
36 views

Jigsaw-style proofs of the Pythagorean theorem with non-square squares

The two squares on the legs of a right triangle can be chopped up (or "dissected") into several pieces that can be reassembled jigsaw-style into a square congruent to that whose side is the ...
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5answers
149 views

What is exactly a “Point”? [duplicate]

I read somewhere that a line is made up of infinite points. Between any two points on that line, there are another infinite points. and between any two points BETWEEN those 2 points there are ...
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1answer
46 views

What is the length of [BC]?

Let A , B and C be 3 points of a circle (c) P is the intersection of two tangents of the circle in points B and C Let (AB)//(CP) and AB=3 and BP=4 What is the length of BC Can someone give hint ! ...
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2answers
73 views

Is this a valid proof that the sum of the angles in a triangle is $180^\circ$?

I think I accidentally found a proof of the famous theorem that the sum of the angles of a triangle add up to $180 ^\circ$, but am not sure if it is correct. Here it is: It can be proved that the ...
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1answer
10 views

If the Lemoine point of triangle ABC lies on the altitude from vertex C, show that either AC = BC or angle C =90 degrees.

There are so many aspects to this problem - I constructed it, but getting the Lemoine Point alone is quite a mess - that I'm not sure where to start with the proof. What aspects of the Lemoine Point ...
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14 views

If angle c=90 degrees in triangle ABC, show that the Lemoine point of the triangle lies on the altitude from vertex C.

I constructed it but it's a bit of a mess with all of the segments and constructions. How exactly would I begin to go about proving that this is true. I can wee that it is the case, but I'm not sure ...
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9 views

Show that the only point in the interior of triangle ABC that is its own isogonal conjugate is the incenter.

Also: Find all other points in the plane that are their own isogonal conjugates. It seems quite obvious that in order for an isogonal conjugate to be itself it must also be the incenter, because it ...
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1answer
18 views

Proving that the midponts of an isosceles trapezoid are collinear

I am trying to prove that the midpoints of the parallel sides of an isosceles trapezoid are perpendicular in order to prove that the lines have identical perpendicular bisectors, but I cannot find a ...
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2answers
35 views

Find BC/AB if I lies in the circumcircle of AEF

Let $ABC$ be a right triangle with angle $B=\frac{\pi}{2}$. Let $E$ and $F$ be the midpoint of $AB$ and $AC$ respectively. If $I$ the in centre of $ABC$ lies on the circumcircle of $AEF$, find the ...
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228 views

Must perpendicular (resp. orthogonal) lines meet?

In geometry books in French, there has traditionally been a very clear distinction between 'orthogonal' lines and 'perpendicular' lines in space. Two lines are called perpendicular if they meet at a ...
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0answers
15 views

What does it mean to say that an affine space is essentially a vector space without a distinguishable origin?

I am having trouble understanding affine spaces. I think that I can wrap my head around the concept of a space without a distinguishable origin. However, this is the definition of 'affine space' that ...
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1answer
37 views

How to find the point of intersection

I apologize in advance since I am not a native English speaker and thus I am not sure if I use the correct English terms for my question... I know that 2 lines are "over" the sides of a triangle ABC ...
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1answer
46 views

Show that if $\angle ADB = 60^{\circ}$ then $AA_1 = BB_1$ (and answer whether the converse is true).

In the diagram above we have that $AA_1$ and $BB_1$ are altitudes and $\angle ADB = 60^{\circ}$. The problem is two fold- show that from$\angle ADB = 60^{\circ}$ it follows that $AA_1$ = $BB_1$ and ...
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1answer
38 views

What is the spherical law of cosines in high dimensions?

If I have 3 points on an N-D sphere, does there exist a law of cosines which holds, regardless of the number of dimensions that the sphere occupies? If so - what is the N-D law of cosines? EDIT for ...
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1answer
27 views

Image of a circle under conformal map $1/z$

The image of a circle under conformal map $1/z$ should be a circle, but how to prove it (or how to find the relationship between the two circles)? $z = x + iy = d + a\exp(i\theta)$, where $a$ is the ...
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1answer
67 views

Locus of centres of circles tangent to two fixed circles?

Find the locus of the centres of circles tangent to two fixed circles. From my initial observations, I strongly think that the locus may be part of a hyperbola or some other conic? (because the ...
2
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1answer
12 views

Saturating space so that at least two lines are close enough

All lines in what follows pass through the origin. The only reason for the angle $2\pi/3$ below is that this is how I began wondering about these questions. Picture the unit disc $S^2$, by which I ...
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2answers
41 views

Prove parallel line is tangent to second circle

Two circles $\Gamma_1,\Gamma_2$ have centers $O_1,O_2$. Let $\Gamma_1\cap\Gamma_2=A,B$, with $A\neq B$. An arbitrary line through $B$ intersects $\Gamma_1$ at $C$ and $\Gamma_2$ at $D$. The tangents ...
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1answer
29 views

Descartes' trisection of angle - getting the cubic equation

I am trying to understand Descartes' (original) proof of trisecting an angle. In the first part of the proof he tries to get a cubic equation, but I cannot really follow his argument. Can somebody ...
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1answer
38 views

Euclidean Distance in 4th Dimension

I have found this question in one of my Universities old pass papers and I'm trying to solve this: You have a fishing rod of length 2 and need to ship it in a box which sides are not longer than 1. ...
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2answers
36 views

Map between circles

Let $C_1, C_2 \subset \mathbb{R}^2$ be concentric circles in the plane. Suppose that $C_1$ bounds $C_2$. Let $f: C_1 \rightarrow C_2$ be a map such that for some $y \in C_2$, $f(x) = y$ for all $x \in ...
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1answer
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Continuous map between circles

Let $C, C_1, C_2 \subset \mathbb{R}^2$ be circles such that $C_1 \cap C_2 = \emptyset$ and $C_1$ does not lie in the interior of the disk bounded by $C_2$ and similarly for $C_2$. Let $f: C ...
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38 views

Constructible decimals?

So I have to figure out if $1.23456$ is constructible. I think that it's not constructible since: I know that this is $\frac{123456}{100000}$ so this goes into $\frac{2^6*3*643}{2^55^5}$ and the ...
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10 views

Point set in affine euclidian planes

Let $\cal{P}$ be an affine euclidian plane, $F_1$ and $F_2$ two points of $\cal{P}$. We consider the following set: $\cal{H}$ = $\{M \in \mathcal{P} \ |\ |MF_1 - MF_2| = F_1F_2\}$ I need to ...
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1answer
16 views

Quadrilaterals and figures

$ABC$ is a triangle in which $L$ is the midpoint of $AB$ and $N$ is a point on $AC$ such that $AN =2CN$. A line thought $L$ parallel to $BN$ meets $AC$ at $M$ prove that $AM=CN$.
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31 views

Prove that there exists an isometry P in Isom(H^2) such that P(A)=B

Where A and B are ideal triangles in H^2 (upper sheet of hyperboloid). How do I get started with this proof?
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3answers
32 views

Check if disk lies within an ellipse

I have an ellipse in normal form centered at the origin and want to check whether a disk with given center point and radius is contained completely in the ellipse without touching it. If I could ...
2
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1answer
77 views

Finding the angles of $\triangle ABC$

In a $\triangle ABC$, from vertex $C$, the median to $AB$, the angle bisector of $\angle BCA$ and the perpendicular to $AB$ divides angle $\angle BCA$ into four equal parts. The task is to compute ...