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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Proving the orthocenter, circumcenter and centroid of a triangle are collinear.

I just had a problem on a test related to Euler’s Line theorem, (Euler Line Theorem: The orthocenter H, the circumcenter O, and the centroid G of any triangle are collinear. Furthermore, G is between H and O (unless the triangle is equilateral, in…
T.Jenkins
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A continuous area-preserving mapping is an isometry?

Suppose that $f\colon\mathbb R^2\to\mathbb R^2$ is a continuous map which preserves area in the Euclidean sense. Can we say that $f$ is an isometry? Note. We donot assume that $f$ is differentiable.
Wall Great
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Proof of Newton's theorem for tangential quadrilaterals

Newton's Theorem for tangential quadrilaterals is this: The center of the circle inscribed into a quadrilateral lies on the line joining the midpoints of the latter's diagonals. For more information, see Cut-the-Knot's entry. Is there any…
RopuToran
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Computing the length of one side of a triangle from distance between orthocenter and the midpoint of another side

$\triangle{ABC}$ is an acute triangle, $\overline{AB} > \overline{BC}$, and $M$ is the midpoint of $\overline{AC}$. $H$ is the orthocenter of the triangle, and $P$ is the point on the minor arc $BC$ that is the intersection of the line through $H$…
user74973
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A geometry problem with an equilateral triangle from a nationat contest 2017

Given an equilateral triangle $ABC$ and a point $O$ on the plain of $\triangle ABC$, such that $\angle AOC=90^{\circ}, \angle BOC=75^{\circ}$. Find the angles of the triangle constructed from segments $AO,BO,CO$. I have proof with trigonometry, but…
user173628192
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Right-angled triangles with integer sides a,b.c,less than 100 such that:c*c=a*a+b*b

$$5*5=3*3+4*4$$ Integers a,b,c For a=1 to 100 For b=1 to 100 For c=1 to 100 if c*c=a*a+b*b list a.b,c Next c Next b Next a I am 74. I am interested in youth-hood practical mathematics. Child can use strings of 3,4,5 to draw a right-angled…
wengweng weng
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Find the planes that are parallel to a line and have a distance of 2 to the point P(1,1,1)

I have to give the equations of the planes that go through the point $(3,4,5)$ that are parallel to the lines with direction $(0,2,1)$ and that are on a distance 2 from the point $(1,1,1)$. So if you say that the other vector of this plane is given…
Mee98
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Alternative proof that all inscribed angles with the same arc are congruent

There is a classic proof that because two inscribed angles with the same arc share the same central angle that is twice the measure of them both, and by the transitive property the two angles are congruent, but I was wondering if there is a way of…
pavle
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How I construct a segment $a^2$?

The length of a segment is given. How can we construct a segment equal to the square of the given segment?
Maxim Kasnedelchev
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Proving that if the Simson line of a triangle goes through the orthocenter, then the pole must be at a vertex.

I’ve already proven that if the Simson line goes through its pole then the pole is at a vertex. Similarly, I’ve proven that if it is perpendicular to one of its sides then the pole must again be at a vertex. How can I prove this for when it goes…
Alex D
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Proving two lines through the circumcircle are parallel.

I'm having trouble solving the following geometry problem and would appreciate any help. I ended up proving some other two lines were parallel instead of the desired ones. Please feel free to change the title to a more descriptive one, as I didn't…
Alex D
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Computing area of a region bound by a triangle and two of its cevians via Menelaus Theorem

$AD$ and $BE$ are two cevians of $\triangle{ABC}$, and $P$ is the intersection of them. If the area of $\triangle\mathit{ABP}$ is 6, the area of $\triangle\mathit{AEP}$ is 3, and the area of $\triangle\mathit{BDP}$ is 4, compute the area enclosed by…
user232552
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Determining the radius of a circle knowing an arc length and a distance

Here's the problem: ($2,5$ is $2.5$) To determine $r$, I used Pythagoras and trigonometry to find that: $\angle{BOC}=\dfrac{\beta}{r}$ $\tan{\dfrac{\beta}{r}}=-\dfrac{\sqrt{(\alpha-r)^2-r^2}}{r}$ As, from the graphic,…
Scientifica
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Triangle Construction By In-Center, Ex-Center and foot of an Altitude

I am struggling to show that constructed triangle is indeed the desired one Condition: In triangle $ABC$, one has marked the in-center, the foot of altitude from vertex $C$ and the center of the ex-circle tangent to side $AB$. After this, the…
Andrew
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Suppose $AB/AF=CB/CD=3$ Prove that $E$ is the midpoint of $\overline{AC}$

Suppose $AB/AF=CB/CD=3$ and the three cevians $\overline{AD},\overline{BE},\overline{CF}$ are concurrent. Prove that $E$ is the midpoint of $\overline{AC}$ I know by cevas theorem that since the cevians are concurrent then $AF/FB \cdot BD/DB \cdot…
HighSchool15
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