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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On inscribed quadrilateral

For $i=1,2$, let: $\Gamma_i$ two circles intersecting each other at $A,B$. $r$ a line containing $A$ intersecting $\Gamma_i$ at $T_i\neq A$. $t_i$ tangent line to $\Gamma_i$ at $T_i$. $P=t_1\cap t_2$. Prove that the quadrilateral $PT_1BT_2$ is…
Sigur
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Bisectors problem

I need help with this geometric problem. Given triangle ABC. CM is the bisector of $\angle ACB, M\in AB$ and $CN, N\in AB$ is the bisector of the suplementary angle of $\angle ACB$. B is between M and N. A circle with diameter MN is drawn. For any…
chen h.
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Calculating weighted Euclidean distance with given weights

This question is regarding the weighted Euclidean distance. I have three features and I am using it as three dimensions. I need to place 2 projects named A and B in this 3 dimensional space and measure the distance among them. But the case is I…
Hiru
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Incentre and excentre of a triangle

Prove that the triangle formed by the points of contact of the sides of a given triangle with the excircles corresponding to these sides is equivalent to the triangle formed by the points of contact of the sides of the triangle with the inscribed…
Apoorv Jain
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Doubts about locus and its equation

Two points A and B with $(1,1)$ and $(-2,3)$ respectively are given.find the locus of point P.So that area of $\Delta$PAB is 9 square units. answer is :- $2x+3y+13=0$ or $2x+3y-23=0$. how i tried:- i assigned unknown point to be P$(x,y)$ and…
Nikhil01
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Rotation of an equation

Problem: Suppose a line $L$ is given by the equation $\frac{x}{a} + \frac{y}{b}=1$, where $a$ and $b$ are non-zero real numbers. Let $\Re_{\frac{\pi}{2}}$ be the counterclockwise rotation of the plane by $90$ $degrees$ with the center at $(0,0)$.…
Michelle
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What do you call a projection of a hyperplane into a finite hypercube that keeps paraxial lines straight?

Similar to the Poincaré disc for hyperbolic space, is there a bijection from $\mathbb{R}^n$ into, say, $[-1,1]^n$, while any paraxial orthotope in $\mathbb{R}^n$ remains a paraxial orthotope after the projection, i.e. lengths are distorted but not…
Pink Bunny
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Properties of sphere

Let $C$ be a circle with diameter $\overline{AB}$. Then it is well known that for any $P$ on the circle $C$ the angle $\angle APB =\frac \pi 2$. There are similar results for sphere?
Chung. J
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How to compute an angle in specific counter-clockwise direction between vectors

I have one incoming vector and multiple outgoing vectors in 2D. I need compute an angle in this way: Imagine an incoming vector parallel to the x-axis. Then the angle-value "starts" below the incoming vector and increases counter-clockwise. Please…
Peter K.
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When two triangles have the same orthocenter and circumscribing circle, are nine points are the same too?

When two triangles have the same orthocenter and circumscribing circle, are the nine points are the same too? If two triangles have the same circumscribing circle, at least the sides have the same length? Could you please give me some explanation…
Student
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we need to show $Ar(\Delta APD)=Ar(ABCD)$

$ABCD$ is a quadrilateral. A line through $D$ parallel to $AC$ meets $BC$ produced at $P$ we need to show $$Area(\Delta APD)=Area(ABCD)$$ I tried but did not get properly. Thank you for helping.
Myshkin
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Triangle inequality for angles

For points $O,A,B,C$ in $\mathbb{R}^{3}$, I was trying to show $\angle AOC \le \angle AOB +\angle BOC$. I could show this when all angles were acute. First, I set $O$ to be the origin and $A,C$ to be on the first quadrant of $XY$ plane and let…
YD55
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Why did Euclid propose in a round-about manner?

For example in Book I. Proposition 2 he shows a line between points B and C. He also shows point A somewhere in the vicinity and shows how one would go about recreating that line starting at point A. Now I would just take my compass, put one end on…
Anthony
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Given three points, can one construct a hyperbolic curve using classical geometric construction method?

Using straight edge and compass method, can a hyperbolic curve be drawn through three given points?
Jeremy Lynes
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Proof with congruence of angles

I came across a proof exercise from my proof work-book that I am stuck on. The questions says: Suppose we have angle PQR with P, Q, and R non-collinear, and ray QS distinct from ray QR such that angle PQS is congruent to angle PQR. Prove that…
Michael Ferashire Silva
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