# Questions tagged [euclidean-geometry]

Geometry assuming the parallel postulate: 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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### Recommend book on elementary geometry

I am seeking recommendations for books on elementary geometry, including Euclidean geometry and analytic geometry.
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### The altitude of an isosceles trapezoid; given area and diagonal

An isosceles trapezoid is given with diagonal $25$ $cm$ and area $300$ $cm^2.$ Find the altitude and the midsegment. Let $DD_1$ and $CC_1$ be the altitudes of the trapezoid through $D$ and $C,$ ...
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### Is there a citation or (at least a date) to ascribe to Conway's Circle Theorem?

There are lots of John Conway tributes (via cardcolm.org) appearing online, and one of the things that I see a lot is the fact that his Circle Theorem (via MathWorld) is a geometric construction that ...
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### Ratio of the heights of triangle is given, determine the sides of triangle.

Can someone help me with this excersize? The ratio of heights to the sides of the triangle is v_{a}:v_{b}:v_{c}=12:5:8. What are the length of the sides of triangle? (a,b,c=?) Thank you! I tried ...
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### $3(a+{1\over a}) = 4(b+{1\over b}) = 5(c+{1\over c})$ and $ab+bc+ca=1$

This was the question : $3(a+{1\over a}) = 4(b+{1\over b}) = 5(c+{1\over c})$ where $a,b,c$ are positive number and $ab+bc+ca=1$ , $5({1-a^2\over 1+a^2}+{1-b^2\over 1+b^2}+{1-c^2\over 1+c^2}) = ?$ ...
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### How to get the left and right rotation quaternions between two $\mathbb{R}^4$ unit vectors?

This question is somewhat similar to this, but in higher dimension. I have two non-collinear unit vectors in $\mathbb{R}^4$, $\bf a$ and $\bf b$. I know from Wikipedia that there is at least one pair ...
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### Trisecting an angle with a compass and 2 marks on a ruler

It's well known that there is no possibility to trisect and angle with a compass and a ruler. But there is such a procedure when the ruler has 2 marks on it. See the snippet below. It's not very ...
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### The base of an isosceles triangle; given leg and radius of circumcircle

An isosceles triangle $\triangle ABC$ is given with leg $AC=5$ and $R=\dfrac{25}{6}$ of the circumcircle. Find the base of the triangle. This was my first sketch. The triangles $AHC$ and $OMC$ are ...
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### Express a vector in terms of a and b and find the angle

The following figure shows an equilateral triangle OAB with side length h. C is the mid-point of AB. D is a point on OA such that OD:DA = 2:1. OC intersects BD at E. OBFE forms a parallelogram. Let ...
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### The leg of an isosceles triangle; given base and area

The area of an isosceles triangle $\triangle ABC$ with base $AB=2c$ is $S.$ Find the leg of the triangle. Let $AC=BC$ and $CH$ and $BP$ be the altitudes through $C$ and $B$, respectively. I am not ...
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### Inscribe a triangle in the given triangle

In a given triangle ABC inscribe the triangle A'B'C' whose sides are parallel with three given lines p ,q ,r ( lines are given as sides of a triangle).
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### Prove that $BQ$ bisects segment $EF$.

Question: Let $ABC$ be an acute angled triangle with orthocentre $H$ and circumcircle $\Gamma$. Let $BH$ intersect $AC$ at $E$, and let $CH$ intersect $AB$ at $F$. Let $AH$ intersect $\Gamma$ again at ...
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### Is it possible to prove the midpoint theorem with just using alternate-interior(exterior) or corresponding angles?

Here is the statement : Let $ABC$ a triangle, $I$ is the midpoint of $[AC]$ and $J$ is the midpoint of $[BC]$. Then the lines $(IJ)$ and $(BC)$ are parallels and $2IJ = BC$. So is it possible to ...
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This proof cannot be found in cut-the-knot.org nor in Loomis' collection. Let $\triangle{ABC}$ be a right-triangle with $\angle{ACB}=90^\circ$. Let $D$, $E$ and $F$ be the contact points of the ...
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### The base of an isosceles triangle; given the radius of the inscribed circle and perimeter

On the sketch below $AC=CB$ and $OD=\dfrac{4}{10}CD$. If the perimeter of $\triangle ABC$ is $P_{\triangle ABC}=40$, find the length of the base $AB$. Let $AC=BC=a$ and $AB=c$. Since the triangle is ...
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### How to verify that matrix is a correct Euclidean distance matrix in $k$-d space?

Given symmetric non-negative matrix $D \in \mathbb{R}^{n\times n}$, how to verify that we can place points $x_1,\ldots x_n$ in $k$-d space such that $D_{ij}=\|x_i - x_j\|_2$? If there is no general ...
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### Does connecting any two points in a graph result in a convex set? [closed]

This is a follow-up question. Let $F:[0,1] \to [0,\infty)$ be a continuous function, and let $G=\{ (x,F(x))\,|\, x \in [0,1] \}$ be the graph of $F$. Is $\cup_{(x,y)\in G^2} [x,y]$ convex?
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### Does connecting any two points in a set result in a convex set?

This might be silly, but I am not sure. Let $A \subseteq \mathbb R^2$. Suppose that for any two points $x,y \in A$, I "add" the straight segment $[x,y]$ between them. Is the result convex? That is, ...