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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Hard cyclic quadrilateral related question

$AB$ is a chord of a circle, and the tangents at $A$ and $B$ meet at $C$. If $P$ is any point on the circle and $PL$, $PM$, $PN$ are the perpendiculars from $P$ to $AB$, $BC$, $CA$, then prove that $$|PL|^2 = |PM| \cdot |PN|$$ I am expected to…
Mathematical Curiosity
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Stuck on a Geometry Problem (II)

(Looking for a geometric solution.) $ABCD$ is quadrilateral, where $|AB|=|AD|=|DC|$. $AED$ is a triangle, where $E$ is the midpoint of $BC$. $\angle DAB=84 ^{\circ}$ and $\angle CDA=72 ^{\circ}$. Determine the measure of $\angle AED$.
blackened
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Drawing an isosceles triangle with Cevians from the base

I would like to draw an isosceles triangle with certain conditions. $\triangle{ABC}$ is an isosceles triangle, and $AB$ is its base. From the endpoints of the base, cevians $AD$ and $BE$ are drawn. They partition the region bound by $\triangle{ABC}$…
A gal named Desire
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Is there a relation between Hamming distance and Euclidean distance?

Is there a relation between Hamming distance and Euclidean distance? If components of vectors are $\in \{1,2\}$,for example. For $\{0,1\}$ everything is obvious.
nnuttertools
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Thales theorem - using euclidien axioms

I wonder if it possible to prove the Thales theorem only with Euclidean axioms. what do you think?
letisya
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Doesn’t a triangle center have to be inside of the triangle?

A ‘Tarry point’, which is on the circumcircle of the given triangle, is described as a ‘triangle center’ by Wikipedia in its article on the Steiner point: “The Tarry point is a triangle center and it is designated as the center X(98) in Encyclopedia…
user584285
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A special case of the Fermat-Torricelli point in a triangle

$\triangle{ABC}$ is an isosceles right triangle; its legs are of length $s = 30\sqrt{5}$ and hypotenuse is of length $30\sqrt{10}$. The Fermat-Torricelli point $P$ must be along the median from the vertex $C$ of the right angle of the…
Adelyn
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Intersection of a circle and a line with only compasses

Recently I asked a question about Mohr-Mascheroni theorem. I read the paper "A short elementary proof of the Mohr-Mascheroni Theorem" by Norbert Hungerbuhler but was unsatisfied with it. In Construction 1 the author constructs intersection points…
Evgeny Kuznetsov
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An optimization problem in Euclidean Geometry - finding "the smallest" inscribed triangle

What is the smallest perimeter of a triangle that can be inscribed in a triangle with sides of lengths $5$, $9$, and $10$? I have read that such a triangle has its vertices at the feet of altitudes from the three vertices of the given triangle. (I…
A gal named Desire
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Prove MB bisects angle IMF

$\Delta ABC$ inscribed $(O)$, incenter $I$, $(I)$ touch $BC$ at $D$. $AD \cap (O)=${$A;E$}. Let $M$ be the midpoint of $BC$ and $N$ be the midpoint of arc $BAC$. Let $EN$ intersect $(BIC)$ at $G$ ($G$ lies in $ABC$). $(AGE) \cap (BIC)=${$G;F$}.…
RopuToran
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Explain why vertical angles must be congruent

I know why vertical angles are congruent but I dont know why they must be congruent
Maximiliano
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General test to prove a set of points creates a cyclic polygon

For $n>4$, is there a way to show that a set of points creates an $n$-sided, cyclic polygon? Or simply, that the set of points are concyclic? All triangles are cyclic. For quadrilaterals, we can use Ptolemy's theorem about the product of…
John Glenn
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Prove GH always passes through a fixed point

Given a fixed triangle $ABC$. A circle pass through $B,C$ meets $AC,AB$ at $D,E$ respectively.$BD$ meet $CE$ at $F$. Let $H,G$ be the projections of $F$ on the internal and external bisector of angle $A$. Prove line $GH$ always passes through a…
RopuToran
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Given a circle and its center, construct the vertices of an inscribed square using only a compass

It is very easy to construct one of the squares inscribed in a circle with the help of a ruler and a compass. My question: Given a circle with its center. Construct the vertices of one of the squares inscribed in this circle using ONLY compass! I…
Peter Boss
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Euclidean K-Center Problem

My google searches have brought me to rather long papers explaining the Euclidean K-Center problem. Can someone please provide a high-level explanation? Thanks.
Kevin Meredith
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