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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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Does this theorem has been proved yet?

I am having a little trouble to prove an hypothesis about Euclidean geometry, This happen because I am using an theorem about the theme that I don't know if is true, can anyone help me? Here is the fount of my problems: If we take four distinct…
sasinhe
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2.6 Drop a perpendicular

Euclidea 2.6 Drop a perpendicular. The Thales theory allow one to construct a right angle at point "a" it seems. Not on the segment. My solution which got 2 stars is construct line from "a" to random point "b" on the segment. Bisect this line at "c"…
Tim Baxter
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Relation of angles with a hexagonal prism intersecting a plane?

If a regular hexagonal prism instersects a plane at an angle, what will be the special properties of the hexagonal cross-section? In particular is there any relationship between the angles? You can't create any hexagonal shape this way, only certain…
zooby
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How to draw a rhombus with 45° angle with minimum movement?

A segment of a line is given and a rhombus with 45° angle is told to construct within 7 elementary moves. Circle and lines are elementary moves.
Anik Paul
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Rotate a triangle to face a fixed point

my problem consists of rotating a triangle around the T axis so that the point A faces the point P. The only known points are A, B, C and the T (the center of the triangle), look at the left figure below. My question is how to make this…
ganzo db
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Construct a circle tangent to given circle and tangent to a given line at a given point.

https://i.stack.imgur.com/EEVNU.jpg Please refer to link for image; given circle at center A, and line h-D, construct a circle tangent to the circle and tangent to line at point D. I have already done another problem similar to it but instead of…
tighten
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Determine if a point is inside a cone generated by three vectors

Assume three vectors $$ \vec{A} = \left(\begin{array}{c} a_x\\ a_y\\ a_z \end{array}\right), \vec{B} = \left(\begin{array}{c} b_x\\ b_y\\ b_z \end{array}\right), \vec{C} = \left(\begin{array}{c} c_x\\ c_y\\ c_z \end{array}\right). $$ How can I find…
Adad
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Limits of Incommensurable quantities

By taking the unit of measure sufficiently small, however, the ratio of two incommensurable quantities can be expressed as nearly to the true value as may be desired. Thus an approximate value of the ratio can be found which shall differ from the…
Omicron
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Can someone explain why this statement is false?

Can someone explain why this statement is false? Although 2000 years of efforts to prove the parallel postulate as a theorem in neutral geometry have been unsuccessful, it is still possible that someday some genius will succeed in proving it.
Wangpin Lin
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Olympiad geometry question.

Let $ABC$ be an equilateral triangle. $D$ is on $BC$ and $BAD$ angle is $20^{\circ}$. Also let $I_1$, $I_2$ be the inner centers of triangle $ABD$, $ACD$ respectively. $E$ is a point making the triangle $I_1I_2B$ be equilateral ($D$ and $E$ are…
EUNSUNG LIM
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Ambiguity regarding numbering and naming of octants

I've only started studying $3D$ geometry and hence came about octants. I tried combining sources online, but there seems to be a lot of ambiguity. My maths book labels the coord axes; and numbers and names the octants in a certain order, the rules…
Gaurang Tandon
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ball Solid geometry

Can you Please help me with finding solution. Thank you! The ball intersects by a plane, whose diameter from the center of the ball is seen as 120 degrees. Find in what relation this plane intersects the volume. And find relation, what respect this…
Garry Boch
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Problem about angles

In triangle $ABC$, prove that the angle between the bisector of $A$ (call $AD$) and the height $AH$ is: $$\measuredangle HAD=\frac{|\measuredangle B-\measuredangle C|}{2}$$ The book states that by writing the equations of different angles, problem…
user325789
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Minimum Number of Distinct Distances

Given that the upper bound on the number of unit distances determined by n points in a real space is $ cn^\frac{4}{3} $ Give a lower bound on the min. number of unique distances. I am really struggling with this, any thoughts appreciated.…
Navy Seal
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Is the KNN classifier, assuming Euclidean distance is used, location-scale invariant? Why or why not?

One question to consider when constructing classifiers is whether or not the method is location-scale invariant. This property holds if subjecting any subset of features in the training data to a linear trans- formation cannot change the prediction.…
Lyu Keting
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