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Questions tagged [euclidean-geometry]

For questions on geometry assuming Euclid's parallel postulate.

The geometry of Euclid is based on five axioms (Euclid called them postulates). Any geometry based on the first four of these is called an absolute geometry. The fifth one states:

If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

It was observed by Proclus that, in the presence of the the other four postulates, Euclid's fifth postulate can be replaced by Playfair's axiom:

Given a line and a point not on it, then one and only one line parallel to the given line can be drawn through the point.

The independence of the parallel postulate and its equivalent formulations from the first four axioms was shown by Beltrami in 1868.

Another alternative definition is that two lines are parallel if every perpendicular extended from one meets the other as a perpendicular.

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Geometric problem about regular triangle

We have a regular triangle $ABC$, where side is $15$, point $N$ belongs to $AB$ such that $AN = 5$. Point $M$ belongs to $AC$ such that $AM = 3$. Prove that $BM$ is perpendicular to $CN$. I tried to make similar triangles carrying the $BK$ segment,…
Zhenya Karapetyan
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How to find the angle made by isosceles triangles on the interior of a kite?

The problem is as follows: In the figure. Find $x$. The choices given in my book are: $\begin{array}{cc} 1.16^{\circ}\\ 2.18^{\circ}\\ 3.20^{\circ}\\ 4.\frac{45}{2}^{\circ}\\ 5.15^{\circ}\\ \end{array}$ According to the official answers sheet the…
Chris Steinbeck Bell
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If $a$ and $b$ both closer to $x$ than $y$, prove that all points on line between $a$ and $b$ are closer to $x$ than $y$

For points $\mathbf{a}, \mathbf{b}, \mathbf{x}, \mathbf{y} \in \mathbb{R}^d$ for some natural $d \ge 1$. I want to prove that if (Euclidean distance) $||\mathbf{x} - \mathbf{a}|| \le ||\mathbf{y} - \mathbf{a}||$ and $||\mathbf{x} - \mathbf{b}|| \le…
dinodeep
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How to find $\frac{AC}{CE}-\frac{BD}{DF}$ given $\frac{AC}{AE}+\frac{DF}{BF}=1$?

The problem is as follows: Let $\textrm{A, B, C, D, E and F}$ collinear points and consecutive. It is known that, $$\frac{AC}{AE}+\frac{DF}{BF}=1$$ Find the value of, $$\frac{AC}{CE}-\frac{BD}{DF}$$ The choices given…
Chris Steinbeck Bell
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Find missing value - geometry

We are given square $ABCD$ and equilateral triangle $CEF$. We are looking for angle $a$. I have tried everything but no clue. All I managed to find is that $EC=EA$ because $AC$ and $DB$ are the square's diagonals, which are perpendicular and bisect…
Harry Gartner
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Geometry problem without trigonometry

In the figure, $AB \bot CK$, $\angle B = 2\angle A$, $I$ and $J$ are the mid-points of $AB$ and $BC$. Prove that $IK = \frac{1}{2} BC$. It can easily be solved with trigonometry by letting $\angle CAB = \theta, BC = \sin \theta$ and show that…
Y.T.
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Lines that cut 2-d object into equal parts, from Zach Star video

In this interesting video from Zach Star https://www.youtube.com/watch?v=IJumRmwYsN4, he falsify the following: False Claim every 2-d shape (open connected) has a point with the property that every line passing throught it, divide the shape in two…
Matteo Testa
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Challenge: find angle x in the figure via logical approach

Background: this problem was from social app, but I cannot find original source. This is an interesting problem that I have not seen similar ones. It is a hard problem, cannot be solved with angle chasing. I worked out via analytic geometry…
r ne
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How to find the angle in between two triangles?

The problem is indicated in the figure from below: The alternatives given in my book are as follows: $\begin{array}{ll} 1.&10^{\circ}\\ 2.&15^{\circ}\\ 3.&20^{\circ}\\ 4.&18^{\circ}\\ \end{array}$ What exactly should be done here?. I'm stuck. The…
Chris Steinbeck Bell
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What accounts for the special relationship between Euclidean geometry and other branches of math?

Many times there are problems which are, in a sense "outside of geometry", but are nevertheless amenable to a geometric approach. For example, I may be asked to prove that the ranges of any two affine one dimensional functions must have either one,…
Mark
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Find the radius of three equal circles, with radius $r$, inscribed in a rectangle.

How can we find the radius of the small circles with the same given radius $r$. Three equal circles, with radius $r$, are inscribed in a rectangle in a way only one of them touches the others two, as the figure indicates. The circles centres form an…
Kamal
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Difference between Haversine and Euclidean Distance

I am fairly new to this geo distance. My use case is to find short distances as a person walks. So I will have 2 sets of (lat,lon)s. Now to find the distance I could use Euclidean distance easily. Looks like the distance conversion will be like…
user 923227
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Common tangent to two circles with Ruler and Compass

Given two circles (centers are given) -- one is not contained within the other, the two do not intersect -- how to construct a line that is tangent to both of them? There are four such lines.
Ilonpilaaja
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Problem about angle in isosceles triangle

$ABC$ is an isosceles triangle, $AB=AC$, and $\measuredangle A=20^{\circ}$. For point $M$ on $AB$ and $N$ on $AC$, we have $AM=NC=BC$. Compute the $\measuredangle BMN$. In a similar problem (with only one point on one side) the key step was to…
user325789
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Definition of an angle

I saw two definitions of an angle. Are those equivalent or is another wrong in some axiomatic system? An angle is the union of two rays. An angle is a subset of a plane restricted by two rays. I guess that they are not equivalent as the second…
curious
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