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.

9328 questions
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construction of a chord that is trisected by a point

Find a construction of a chord through a point P such that P divides the chord in the ratio 1:2 in any given circle. Cleary not all points P work, so I'm trying to find the construction when it is possible. What is invariant about all such…
Baker013273213
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Given an angle and a point, construct a circle through the point and tangent to the sides of the angle

Given an angle $\angle ABC$ and a point $P$ inside of it, draw a circle which passed through $P$ and is tangent to both $\overleftrightarrow{AB}$ and $\overleftrightarrow{BC}$. Well, the center has to lie on the angle bisector (obvious). The…
Baker013273213
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Generalizations of Mordell's theorem on rational quadrilaterals

Mordell proved that every quadrilateral in the Euclidean plane is arbitrarily close to a quadrilateral whose sides and diagonals are rational. What if we take five points in the plane? Is there a choice of 5 bad points so that for some positive…
Peter Kropholler
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Is there a word for rotating and scaling an object at the same time?

I consider this a mathematics questions because I am building a library that allows perspectival distortions projected over a map. here is an example of the action I am describing The action is built with math formulas written for both the scale and…
devdev_dev
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CE is independant to choice of CAB why?

If points A , B , C are fixed and D is intersection point of two lines ( Angle bisector of A and Perpendicular bisector of BC ) let E is point on line AC such that line ED is perpendicular to line AC Then length of segment CE is determined by…
Solvable Potato
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Cross-ratio concept from Euclidean Geometry

I have seen on the following Wikipedia website the definition of "cross-product." Apparently, if four lines passing through a common point $P$ are traversed by a line and the points of intersection "in one direction" are A, B, C, and D, their…
A gal named Desire
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How to find the area of a kite from a given area of triangle

In a kite $AEDC$, the shaded area is known, and $EB \parallel DC$. It is also given that $3|AB|=4|BC|$. How can one find the area of the kite? I have tried using similar triangles (drawing the diagonals $AD$ and $EC$), since the ratio between $AB$…
Ido Sarig
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Sliding a semicircle within a rectangle

Let ABCD be a rectangle with sides sizes of $1$ and $CD=AB=\sqrt{2}$. And let a semicircle with center $O$ of radius $1$ be such that its diameter is on the $AB$-produced so $CD$ will be tangent to the circle at point $Q$. $AB$ is always within the…
user200918
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A geometric problem for a quadrilateral

1) I have to calculate the area of a kite $$ABCD$$ with $$AB-CD=(\sqrt {2}+1)(\sqrt{3}+1)$$ and $$11 \angle A= \angle C.$$ 2) A second question is that if $$2 AB^2+AC^2+2 AD^2=4 BD^2$$ then there is $$ \angle A= 4\angle C.$$ My attempt: I think…
George
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Thales Application

I'm struggling with this exercice: Let $ABCD$ be a trapezium. $AB$ and $CD$ intersect at $O$. $OE \parallel BD$, $OF \parallel AC $ and $F,C,B,E$ are aligned points Prove that $EB=CF$ I think it's an application of Thales' theorem. I applied the…
Rfgauss
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Related to angles in tetrahedron

Let $OABC$ be a tetrahedron such that $|OA|=|OB|=|OC|$. Denote by $D$ and $E$ the midpoints of segments $AB$ and $AC$ respectively. If $\alpha=\angle(DOE)$ and $\beta=\angle(BOC)$ what is the ratio $\beta/\alpha$? It is obvious that $|BC|=2|DE|$…
Arian
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Are perpendicular bisectors of an arbitrary $n$-gon concurrent ? If so, how to prove it?

Are perpendicular bisectors of an arbitrary $n$-gon concurrent ? If so, how to prove it ?
Dr. user44690
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Prove that every circle passing through a fixed point and having its centre on a fixed straight line must pass through another fixed point

Prove that every circle passing through a fixed point and having its centre on a fixed straight line must pass through another fixed point Please help me understand the question and provide me with a geometric proof
Rishabh Shetty
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Minkowski sum of a circle and ellipse

What will be the Minkowski sum of a circle and ellipse? Will it be an ellipse of the major axis (a+r) and minor axis (b+r) centered at the location of (x+y), where x is the center of the circle of radius r and y is the center of the ellipse?
Kaushlendra Pandey
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Which double cone splits a sphere into two equal volumes?

Consider a double cone (like a past and future light cone) centered at the origin. Now imagine a sphere centered at the origin. What angle of the slope of the double cone makes it so that the it splits the sphere into 3 pieces such that the…
zooby
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