# Questions tagged [euclidean-geometry]

Geometry assuming the parallel postulate: in a plane, given a line and a point not on that line, there is exactly one line parallel to the given line through the given point.

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### Really really hard and old Euclidean geometry problem

Let $M$ be the midpoint of side $AB$ of triangle $ABC$. Let $P$ be a point on $AB$ between $A$ and $M$, and let $MD$ be drawn parallel to $PC$ and intersecting $BC$ at $D$. If the ratio of the area of ...
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### Distance between points maximally distributed on n-dimensional unit sphere?

This problem arose in some of my own personal data science research and I am wondering if anyone has encountered this before. Consider $k$ points that lie on an $n$-dimensional unit sphere such that ...
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### Sine difference identity

I'm trying to prove that given 2 vectors $\vec{a} = A(\cos{\alpha}, \sin{\alpha})$ and $\vec{b} = B(\cos{\beta}, \sin{\beta})$ the following relation is true by using the exterior product with the ...
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### Two inscribed circles are tangential to a chord with diameter line at $30^\circ$ with chord. Find radius ratio of two circles

I found the problem here (I can't see deleted posts) but the post got downvoted and deleted soon, but I felt so inspired to find the solution that can't let the problem rot by itself, so, rather re-...
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### Difficult problem in elementary euclidean geometry

A point D is chosen inside an equilateral triangle $ABC$ such that $AD$ = $BD$. A point $E$ outside the triangle is chosen such that $\angle DBE$ = $\angle DBC$ and $BE$ = $AB$. Find the degree ...
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### Can I make this assumption?

I am solving this question: Let $ABC$ be an acute angled triangle and $CD$ be the altitude through $C$. If $AB=8$ and $CD=6$, find the distance between the midpoints of BC and AD. So I observed that ...
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### Construct any regular polygon that has the same area as the sum of $n$ given triangles

Original question: Construct any regular (or similar-scaled to a given) geometric shape that has the same area as given triangle? My idea is application of generalized Pythagora's theorem. Euclid ...
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### Generalizing Routh's Theorem to quadrilaterals

Routh's theorem gives the area of a triangle determined by three cevians in a parent triangle, in terms of the "Ceva ratios" for those cevians. After learning about Routh's theorem, I started ...
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### Solve for the area of a parallelogram; given diagonals and a side

Find the area of the parallelogram $ABCD$ with side $AB=10\sqrt{3}$ $cm$ and diagonals $BD=10\sqrt{3}$ $cm$ and $BC=10$ $cm.$ Using the fact that $AC^2+BD^2=2(AB^2+BC^2)$ we can find the other side ...