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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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Prove that the symmetric point of orthocenter $H$ of a triangle with respect to the midpoint of any side resides on circumcircle.

Prove that the symmetric point of orthocenter $H$ of a triangle with respect to the midpoint of any side resides on the triangle's circumcircle. In the above figure,it's sufficient to prove that triangle $BHM$ is equal to triangle $CKM$, but…
Hamid Reza Ebrahimi
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In the following figure prove that: $KO||BM$

$AA' , BB'$ are two perpendicular diameters of circle C(O).Consider arbitrary point $M$ on C between $A'$ and $B'$.Chord $BM$ intersects diameter $AA'$ at $N$.Draw a perpendicular line $d$ to $AA'$ through $N$.If tangent to the circle at $M$…
Hamid Reza Ebrahimi
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Assume that a part of ∆ABC around vertex A is not visible. Describe how to find the angle bisector of ∠CAB.

Assume that a part of ∆ABC around vertex A is not visible. Describe how to find the angle bisector of ∠CAB. I have no idea how to begin this problem. Any help is appreciated. Thank you.
Lily
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Projections of lines in $\mathbb{R}^n$ to the torus $\mathbb{T}^n$

Let $\mathbf{e}\in\mathbb{R}^n$ be a unit vector, $\hat L:=\mathbb{R}\mathbf{e}$ be the line in the direction of $\mathbf{e}$, which projects down to a line $L$ on the torus $\mathbb{T}^n:=\mathbb{R}^n/\mathbb{Z}^n$. Question 1. Is the following…
Pengfei
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In any triangle $ ABC $ $\frac {sin(B-C)}{sin (B+C)}=\frac {b^2-c^2}{a^2} $.

In any triangle $ ABC $ prove that $\frac {sin(B-C)}{sin (B+C)}=\frac {b^2-c^2}{a^2} $. Please help. Thanks in advance.
D. N.
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Find a point further up on the steepest slope line in 3 dimensions?

I can find the coordinates of point B on the slope line in 2 dimensions. How do I find a similar point on the steepest tangent line in 3 dimensions? Starting with a point A with coordinates $\left(x_0,f(x_0)\right)$ that lies on a function f I can…
Conor
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What do you advise me to do to understand middle school maths?

people. I am very new to stack exchange, and i just came into the maths section (i usually stick to coding section) because maths has started to get harder for me. I am now going to eight grade, and i'm gonna have to learn a lot about new stuff,…
user353031
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Proof - Elementar Geometry (parallelogram)

Prove that by connecting midpoints of adjacent sides of a quadrilateral we get a parallelogram. I'm having problems with this piece of work for some time so decided to ask for help here. Though I'm not sure I translated it purely mathematically so…
Damjan Babić
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Apollonius' circle

A theorem states, that the three Apollonian circles, associated with the given triangle $ABC$ with sidelengths $a \neq b \neq c$ intersect in two points. The proof proceeds by showing that if the two circles passing through $A$ and $B$ intersect at…
Ilonpilaaja
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mod used to describe an angle

Reading Pedoe's "Geometry: A Comprehensiveness Course" I came across the following We know that from Euclidean geometry, for any triangle ABC,$$\sphericalangle ABC + \sphericalangle CAB + \sphericalangle BCA = \pi \quad (\text{mod}\ 2\pi) $$ I am…
Diehardwalnut
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Confusion on wording of an elementary geometry problem

I really want to know the following geometry problem is valid or not. (Please don't change the wording of the problem. Please answer it is valid or not. Please answer frankly.) "ABCD is a parallelogram. Any circle through A and B cuts DA and CB…
Aung Kyaw
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An elementary problem in Euclidean geometry

Let $ABC$ be an acute triangle ($AB < AC$) which is circumscribed by a circle with center $O$. $BE$ and $CF$ are two altitudes and $H$ is the orthocenter of the triangle. Let $M$ be the intersection of $EF$ and $BC$. Now draw the diameter $AK$ of…
Nguyễn Trọng Tài
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How do I make an inner rounded rectangle and an outer rounded rectangle be parallel around the corners?

The outer radius does not follow the inner radius. I am currently using x = width/4 + radius + outset, y = height/4 + radius + outset. I think the outset needs to be some ratio of the hypotenuse.Please view my image on the link. rounded rectangle…
Gordon
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Geometrical proof required regarding midpoints

$ABCD$ is a convex quadrilateral with $W, X, Y, Z, M$ and $N$ as the midpoints of $AB, BC, CD, DA,$ the diagonals $AC$ and $BD$ respectively. [Then, $WXYZ$ is a parallelogram with $K$ as the intersection point of its diagonals by Varignon's…
Mick
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Given a diameter of a circle bisecting the angle formed by two intersecting chords, Prove the chords are equal

In a circle, a diameter bisects the angle formed by two intersecting chords. Prove that the chords are equal
Kevin
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