# Questions tagged [euclidean-geometry]

For questions on geometry assuming Euclid's parallel postulate.

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### concyclicity related to the Humpty point

$AD$ and $CE$ are the altitudes of triangle $ABC$, $M$ is the midpoint of $AC$. Circumcircles $\omega_1$ and $\omega_2$ of triangles $AEM$ and $CDM$, respectively, intersect at point $P$. $CP$ and $AP$...
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### Alternative proof of Apollonius's identity

How does one prove Apollonius's identity without coordinates or the law of cosines (or similar trigonometric laws)? Is there a known rearrangement proof similar to the many such proofs for the ...
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### Conditions on a finite metric that guarantees embedding in Euclidean space? [duplicate]

If we have $n$ points in some metric space, do there exist coordinates for the $n$ points in an $n-1$ dimensional Euclidean space with exactly the same pairwise distances as in the original space? ...
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### Find $\angle BCD$

A convex quadrilateral $ABCD$ satisfies $\angle CAB= \angle ADB=30^\circ,\angle ABD=77^\circ$ and $BC=CD.$ Find $\angle BCD$ My attempt: Constructed the perpendicular bisector of $DB$ to meet $AD$ ...
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### Double Contact Chained Ellipses Problem

A few years ago, when I played around with GeoGebra, I came up with the following conjecture. Conjecture Let $n\in\mathbb{N}, n\geq3$. Let $E$ be an ellipse, and let $E_{1}, E_{2}, \dots, E_{n}$ be ...
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1 vote
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### Properties that relate to the chord of a parabola passing from the perpendicular projection of the focus point on the parabola guide [closed]

Now I remembered my previous question about finding the harmonic mean using a parabola and my answer, which included a second method. That second method inspired me to try more in this configuration ...
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### Find the side size of the square in the figure below

In the plane figure below, we have a parallelogram $ABCD$ and three squares, two of which have side measurements equal to the sides of the parallelogram and the largest of which has side measurements ...
• 7,031
1 vote
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### Alternate Proof for Sum of Sides of a Triangle Inequality

I recently stumbled upon an idea for a proof for the sum of two sides of a triangle inequality. Note that I am just a high school student and feel free to correct me wherever if I am wrong. Statement/...
1 vote
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### Sum of the vectors from centre $O$ to the polygon vertices

I'm attempting to calculate the sum of the vectors from the center of a regular polygon to each of the vertices. I have already solve it in a complex analysis manner: To represent the vertices of a ...
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### Find the size of the radius of the nth circle.

In the figure shown, calculate the size of the radius of the nth circle. (Answer:$\frac{R}{n^2+2}$) I calculated the radius of the first two circles. How to extrapolate this resolution to other rays? ...
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### Triangular inequality on the angles of the faces of a trihedron [duplicate]

I just saw this theorem which I'm trying to prove but got stuck. Given a trihedron like the one below, whose angles between its edges are $\widehat{bc}, \widehat{ac}$ and $\widehat{ab}$, then prove ...
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