Mathematics Stack Exchange
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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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How to find the angle in a triangle when two bisectors cross?

The problem is as follows: The figure from below shows a figure. Find the requested angle $x$ using the information given. The alternatives given in my book are as…
Chris Steinbeck Bell
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How to find the angle in a triangle inside of a quadrilateral?

The problem is as follows: Find the angle $x$ as indicated in the figure from below: The alternatives given in my book are as…
Chris Steinbeck Bell
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Problem with four equal-sized Circles

I think the following is true but haven't managed to prove it yet. Consider a circle with center $O$ and radius $r$. Choose three points $A,B,C$ on or inside the circle such that all sides of $\triangle ABC$ have length greater than $r$. Show that…
user92596
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Is there a non-right triangle whose circumcenter lies on one of its sides?

Here is one elementary, yet interesting geometric question. If we have a right triangle, its circumcenter is the midpoint of its hypotenuse. How to prove other direction? Namely, Is there a non-right triangle which corresponding circumcenter is…
1b3b
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A property of trapezoids - Steiner's Theorem

An important feature of trapezoids is that the midpoints of its bases, the intersection of its diagonals, and the intersection of the lines through its legs are collinear. Explanation for three of these points being collinear $\mathit{ABCD}$ is a…
user74973
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The points $M$ and $N$ are chosen on the angle bisector $AL$ of a $\Delta ABC$ such that $\angle ABM=\angle ACN=23^0$.

Question: The points $M$ and $N$ are chosen on the angle bisector $AL$ of a $\Delta ABC$ such that $\angle ABM=\angle ACN=23^0$. $X$ is a point inside the triangle such that $BX=CX$ and $\angle BXC = 2\angle BML$. Find $\angle MXN$. Solution: Let…
Sanket Biswas
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Euclid's fifth postulate and corresponding angles postulate

The Corresponding Angles Postulate states that, when two parallel lines are cut by a transversal , the resulting corresponding angles are congruent. Is this a direct obvious consequence of Euclid's fifth postulate? I think so, because the fifth…
user761982
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Is it possible to draw an isosceles triangle with a compass and a ruler?

Is it possible to draw an isosceles triangle with a compass and a ruler? The ruler is not marked, the two legs of the compass cannot leave the paper, and the two points on the plane are known. **I mean the two points at the base of the isosceles…
netF
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All Possible Notations of Simple Quadrilaterals Using Permutations

Let $\square ABCD$ be a simple quadrilateral. Now there are many ways to represent the same quadrilateral, such as $\square BCDA$ or $\square ADCB$. These notations can be described by permutations; for example, the following permutation will…
WishofStar
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Rotation around an axis if axis polarity is inverted

Consider a rotation around the Z axis, determined by angle $\theta$. If I change the polarity of the X or Y axis, the rotation becomes $- \theta$, and stays $\theta$ if both are inverted. But what about if I change the polarity of the Z axis itself?…
PaulSiML
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Euclidean cirle question

Let $c_1$ be a circle with center $O$. Let angle $ABC$ be an inscribed angle of the circle $c_1$. i) If $O$ and $B$ are on the same side of the line $AC$, what is the relationship between $\angle ABC$ and $ \angle AOC$? ii) If $O$ and $B$ are on…
Diane Vanderwaif
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Applying Playfair's axiom when the point does not lie on the given line

Playfair's axiom states: Through any point in the plane, there is at most one straight line parallel to a given straight line. This axiom is equivalent to the parallel postulate. Intuitively, we know that if the point $P$ does not lie on the…
Siddhartha
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Proposition 1.28 of Euclid's Elements

Proposition 1.28 states: If a straight line falling on two straight lines makes the sum of the interior angles on the same side equal to two right angles, then the straight lines are parallel to one another. Euclid has given a somewhat long…
Siddhartha
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$[0,1]$ is equidecomposable with $(0,1]$

What decomposition of $[0,1]$ would get rid of the point $0$? I feel like the idea would involve picking a point $u\in [0,1]$ and sending $[0,u]$ to $[1-u,1]$ but that would require sending $(u, 1]$ to $(0,1-u]$ but this decomposition is not…
chittychitty
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Prove triangle with integer sides, area, inradius and circumradius has even sides and P divisible by 4

Triangles with integer sides (a,b,c), perimeter (P), area (A), inradius (r), and circumradius (R) necessarily have sides which are even, and a perimeter divisible by 4 and I'd like to prove this but cannot as yet. I've search the web and this site…
Dominic
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