Mathematics Stack Exchange
Mathematics Stack Exchange

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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Geometry with right triangles

I have this How do I find $h$? I know that I must use the cosine of $70$ degrees but I'm not sure how.
user108343
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How to derive rotation of a point?

Suppose a point has coordinates (x,y) and we rotate it through an angle $\alpha$, now it has new coordinates (X,Y). I know the values of the new coordinates are:- $$X =x\cos\alpha-y\sin\alpha$$ $$Y =x\sin\alpha+y\cos\alpha$$ I was trying to find the…
Abhishek Mhatre
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Line through midpoints of two parallel chords

I heard that in euclidean geometry one has to prove trivial things. I was wondering how one could prove this one? Let $O$ be a center of a circle $\Gamma$. Suppose that $A,B,C,D$ are points on the circumference of $\Gamma$ such that $AB||CD$. Let…
guest
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Show that P is equidistant from B and C.

P is drawn using the exterior angle bisector of A.
Charles
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In triangle ABC, prove that angle A is a right angle if and only if the length of the median from A to BC is exactly half the length of side BC.

I drew it to scale because I was trying to notice something, but I cannot figure anything out.
Charles
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what is the barycenter of a wire tetrahedron?

The barycenter of a wire triangle is the "Spieker" point, namely the center of the inscribed circle of its medial triangle, that is the triangle whose vertices are the midpoints of the sides. Is there a nice caracterisation for the barycenter of a…
mark
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Let P be the point where BY and CZ meet. Assume that BY=CZ and PY=PZ. Show that AB=AC.

I've been able to prove that ΔBPC is isosceles using given information that CZ=BY, but I am at a loss with how to prove that ΔBPZ≅ΔCPY. If I could prove this I could show that AB=AC since the equal base angles would make it an isosceles triangle.
Charles
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Equation of a line about which we are reflecting

Let $A$ be the matrix of a reflection about a line of the euclidean plane (w.r.t. the standard basis). How can I find the equation of the line?
TheWanderer
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Mathematics of photography

From mathematics perspective, cameras do convert the 3d shapes into 2d shapes in the photos. If we consider a 3D coordinate system X-Y-Z which the origins is the camera (or its lens or things like that) and select direction like this: Where Blue:…
epsi1on
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Be $m$ and $n$ two perpendicular lines, and ...

Be $m$ and $n$ two perpendicular lines, and be distinct points $A$ and $B$ outside the lines and in the first quadrant. What is the shortest way to get from point $A$ to point $B$ by tapping the two lines? Idea: Listen say that the shortest path…
Danilo Santana
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Subset convex of plane

A plan of the subset is $convex$ if the segment connecting any two of its points is fully contained therein. The simplest examples of $convex$ $sets$ are the plan itself and any half-plane. Show that the intersection of two semi-planes is a…
Danilo Santana
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Hard problem about law of the cosine

I have been trying without success to prove by contradiction the following problem: Given 5 segments $x_1\leq x_2\leq x_3\leq x_4\leq x_5$ each three of which are sides of a triangle. Prove that there exists an acute angled triangle with sides…
parkhyeyoo
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How to show express $y $ in terms of angle $\theta$?

$ABC$ is a straight line with $AB = BC = 3$ units. $B$ is the centre of the circle with radius of $2$ units. $P$ is a point on the circle. $\widehat{B_1} = \theta$, $\widehat{A} = x$, $QC \perp AC$ and $QC = y$. Show that $$y =…
Aspiring Mathlete
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Given a triangle $ABC$, make it a point $D$ on the side $AB$.

Given a triangle $ABC$, make it a point $D$ on the side $AB$. Show that $\overline {CD}$ is smaller than the length of one of the sides $BC$ and $AC$. Ideas? The triangular inequality will not. I wanted to try the theorem of the exterior angle and…
Danilo Santana
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Two lines intersect forming four angles

Two lines intersect forming four angles. If one of them is right, show that others are too straight. I am clueless how to start. Ideas?
Danilo Santana
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