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 formed by an interior bisector and a suplementary from another bisector?

The problem is as follows: $AD$ bisect angle $\angle BAC$. Point $E$ which lies on $BC$ is generated by the $DE$ which is a bisector of $\angle ADC$. Using this information and the figure from below, find the angle which is made by $DE$ and…
Chris Steinbeck Bell
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How to find the angle formed by a cevian in a right triangle given a condition?

The problem is as follows: In a right triangle $\triangle ABC$, right at $B$, it is traced the height $BH$, in the triangle $BHC$, then is traced the interior cevian $HM$ in such a way that $MC = AB$. Find the measure of the angle $\angle MHC$ if…
Chris Steinbeck Bell
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How to find the angle formed by the intersection of three lines forming a triangle?

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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Finding $AC$ given that $ABCD$ is inscribed in a circle

Suppose quadrilateral $ABCD$ is inscribed in a circle such that $AB=4, BC=5, CD=6,$ and $DA=7.$ Find the length of $AC.$ I do realize that this is a repost, but the previous asker and answerers did not state a clear method as to how to do this…
questionasker
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Ptolemy's theorem generalized for other cyclic polygons

(It's my first time using a math forum for help so please bear with me) I was working on a relatively simple generalization for Ptolemy's theorem to all cyclic polygons, I've verified that the proof works with pentagons and hexagons so I'm…
person
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Can we use Euclid's second postulate ("A terminated line can be produced indefinitely") to say that the universe is infinite?

Euclid's second postulate says A terminated line can be produced indefinitely. Can we use this and say, Universe is infinite? I have read this post but if we consider universe as a three dimensional space, the universe will fall in to the…
Madhusoodan P
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Congruent parts of triangles

How can you show that if two sides of a triangle are not congruent, then the angles opposite those sides are not congruent, and the larger angle is opposite the larger side of the triangle. [clue: Use isosceles triangles.]
caleb
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Show that the locus of midpoints of a family of parallel chords of a circle is a diameter perpendicular those chords.

Show that the locus of the midpoint of a family of parallel chords of a circle is a diameter which is perpendicular to the given family of chords. Please help me understand the question with a figure .
Rishabh Shetty
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find the two circles inscribed by three lines

Let $l$ be the (non vertical) line with equation $y=mx+b$. $l$ determines two circles tangent to $l$ and the two vertical lines: $x=-1$, $x=1$. Clearly, the centres of these circles both lie on the y-axis. What are their y-coordinates?
sitiposit
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Given $2$ points, find the lines passing through the points which have a certain distance

In a $3$-dimensional Euclidean space, we are given the points $A(1,0,-2)$ and $B(3,-1,1)$. Find two lines $a$ and $b$ such that $A\in a$ , $B\in b$ , and the distance $d(a,b)$ between the lines is $2$.
M.M.
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Equality of angles formed by parrallel lines and a transversal - no proof?

At 4:43 of the following video Khan from Khan Academy states that there is no proof that two parallel lines crossed by a transversal line have the same angle. Rather it's to be taken on…
Tony
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prove every pair of points $P,Q, d(P,Q)>0$

Prove: For every pair of points $P, Q$ 1. $d(P,Q)>0$ 2. $d(P,Q) = 0$ if and only if $P=Q$ 3. $d(P,Q) = d(Q,P)$ where $d(P,Q)$ is defined as the distance between $P$ and $Q$ and $d$ is a function For the first part, I tried to assume the contrary,…
user141447
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Problem related "polygon offseting"

Knowing points P1,P2,P3 and distance d and the angles shown in the figure, angle between a and b not necessary 90º What's the size of K?
Gabriel
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Euclid's Elements, Book 1, Prop 2: why the faff?

I propose an alternative: set your compass to BC, draw a circle around A, draw a line from A to any point on the circumference, basta. Why is his better? We're both using postulate 3. We both end up with the new line at a random angle. But I don't…
Adrian May
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Find the point P on the joint of A(1,4) and B(5,6) that is twice as far from A as B is from A

And lies: (i) On the same side of A as B does (ii) On the opposite side of A as B does Now I already know the section formula which is applied when the point is outside the line. I plug in the values and somehow I get different answers. The section…
Arkilo
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