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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Common Intersection Point of ellipsoids

Suppose we have two ellipses in $2$-dimensions centered at the origin. It is easy to visualize that (unless one is contained in the other) they will have $4$ points of intersection. Is it known whether in three dimensions, three generic ellipsoids…
fuzzyrock1
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A right hexagon and right pyramid

Does it possible to obtain a regular hexagon as a section of right pyramid with the base of the form of regular pentagon? O.Ganyushkin
Roman83
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Proving $AE+AP=PD$ In a Certain Right Triangle

$\angle B$ in $ \triangle ABC$ is right. The incircle of $ \triangle ABC$ is tangent to the side $AB,BC,CA$ in $E,D,F$. The line $AD$ meets the incircle of $ \triangle ABC$ in $P(\neq D)$. If $\angle BPC$ is right, prove that $AE+AP=PD$. I have…
Chad Shin
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Pyramid and perpendicular planes

I need help with the following problem. Given a pyramid ABCDM. The opposite planes (ABM) and (DCM) are perpendicular to the base ABCD. The base ABCD is trapezoid (AD || BC) and AD=3 cm., BC=5 cm. If the area of the triangles ABM and BCM is $20 cm^2…
Adam
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Proving that the midponts of an isosceles trapezoid are collinear

I am trying to prove that the midpoints of the parallel sides of an isosceles trapezoid are perpendicular in order to prove that the lines have identical perpendicular bisectors, but I cannot find a way to prove such a thing.
Thomas Greim
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Proof that a secant line intersects a circle in exactly two points (according to Hilbert's axiomatic system)

With Hilbert's axiomatic system, How do I prove that a non-tangent line $d$ that intersects a circle $C$ intersects it in exactly two point? My teacher gave us the following clue: First show that if ABC and A'B'C' are two triangles with right angles…
Mathphilo
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Question on Euclid’s 5 Postulates

Here are the five postulates: Each pair of points can be joined by one and only one straight line segment. Any straight line segment can be indefinitely extended in either direction. There is exactly one circle of any given radius with any given…
ChaoSXDemon
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a question about the angles between the diagonals of a trapezoid.

We have a trapezoid with bases $4$ and $9$, and diagonals $5$ and $12$. Determine the area of it and also the angle between the diagonals. The area is $15\sqrt{2}$; with cosine formula it is $90$ degrees, but I don't want to use it. I have done…
kpax
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Is it possible that Euclid plane can be modelled using countable number of points?

I know that the Pythagorean already showed that the set of all rationals cannot measure all "real" distances, but what if we assume that the number of points and distances in Euclid plane is countable (but not rationals). In this case if we add all…
Yonatan
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Show that XY is parallel to CD

AB is parallel to CD. CD is not a diameter. I want to show that $\triangle ZCD$ is similar to $\triangle ZXY$ but I don't know how to get there. The only thing I had in mind was using the arcs, but that didn't get me anywhere.
Charles
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Given a convex quadrilateral, assume that angle ADB = angle ACB. show that angle ABD= angle ACD

Quit easily I've been able to show that triangle AED is similar to triangle BEC by angle-angle (we're given the two angles, and AED and BEC are vertical angles). I'm absolutely stuck with where to go next. I want to show it using similar triangles.
Charles
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proving that $BC' \parallel B'C$

Given $3$ different points on line $p$ : $A,B,C$, and $3$ different points on line $q$ : $A',B',C'$. In addition: $A'B \parallel AB'$, and $A'C \parallel AC'$. The lines $p$ and $q$ are intersecting at point $M$ which is different from the points…
may
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Let A, B, C and D be placed consecutively on a circle. Let W, X, Y and Z be the midpoints of the arcs AB, BC, CD and DA, respectively.

Show that the chords WY and XZ are perpendicular. I've drawn it using Geogebra and it is quite obvious that it is true - regardless of how I manipulate it, I just don't know where to start with proving it. If I draw line segments WX and YZ I have…
Charles
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Similar triangles

Knowing that Triangle $LAB$ is similar to Triangle $LRQ$, prove that the length of $QR$ is constant while point $L$ varies. There are two circles intersect at points $A$ and $B$. $L$ is a point on first circle that is free to move, whereas $LA$ &…
user31284
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Given that AB=AC and BY=CZ, prove that PY=PZ.

In this image $P$ is the point where $\overline{BY}$ and $\overline{CZ}$ cross. $\Delta ABC$ is isosceles, and proving that $\Delta BPC$ is isosceles will be enough to show that $\overline{PY}=\overline{PZ}$, but I cannot determine a way to show…
Charles
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