Articles of euclidean geometry

Identifying the type of conic using its symmetric matrix in the extended Euclidean plane

I have represented the conic $C$ in the Euclidean plane described by the equation $$3x^{2}+ 2y^{2}+ 7xy + 4x + 5y + 3 = 0$$ as a symmetric matrix. \begin{pmatrix} 3 & 7/2 & 2\\ 7/2 & 2 & 5/2\\ 2 & 5/2 & 3\end{pmatrix}. Now I am asked to show that $C$ is nonsingular […]

Constructing two tangents to the given circle from the point A not on it

I’m trying to complete Level 21 from euclid the game: The goal is to construct two tangents to the given circle from the point A not on it. So far I’ve figured that the segments from B to the tangent points must be equal. And of course the triangles AB[tangent point] are right angles. […]

we need to show $Ar(\Delta APD)=Ar(ABCD)$

$ABCD$ is a quadrilateral. A line through $D$ parallel to $AC$ meets $BC$ produced at $P$ we need to show $$Area(\Delta APD)=Area(ABCD)$$ I tried but did not get properly. Thank you for helping.

Pairs of isometries that jointly fix a set (revised)

Let $E$ be a Euclidean space, and let $X, Y\subseteq E$ be two disjoint subsets. Suppose that there exist isometries $i$ and $j$ such that $i(X)\cap j(Y)=\emptyset$ and $i(X)\cup j(Y)=X\cup Y$. Does it follow that one of the following is true: (i) there’s an isometry fixing $X\cup Y$, mapping $X$ to $i(X)$ and mapping $Y$ […]

Relating area to a line intersecting with a point.

I really could use a hint with this following problem: If a line L separates a parallelogram into two regions of equal areas, then L contains the point of intersection of the diagonals of the parallelogram. The figure shows a line L horizontally through the sides of the parallelogram. This creates two trapezoids and I […]

Circle-Cirle Intersection

Sorry for asking a question like this (can’t comment as my reputation is too low). Could someone please explain this answer to me? Particularly the part, where the orthogonal vectors are declared? Why are the vectors declared as such? What is the reasoning or math behind declaring them in such a way? Given the points […]

How to determine 2D coordinates of points given only pairwise distances

I have a set of 2D points in which each pair has a known Euclidean distance between them. How can I go about determining an arrangement of them? I understand there is not a unique solution in general, but for the sake of my question, assume one point is fixed at the origin. Mathematically, we […]

How to generalize parallelograms to non-Euclidean spaces?

Question: How can one generalize parallelograms to non-Euclidean spaces? In particular, how can one generalize parallelograms to Finsler manifolds which are not necessarily affine spaces (i.e. to “spaces with norms which don’t satisfy the parallelogram law”)? Or at least to Riemannian manifolds which are not affine spaces? The dream of course would be a generalization […]

In an N-dimensional space filled with points, systematically find the closest point to a specified point

This is my first post on the mathematics stack exchange site. If I am posting this question on the wrong site, I apologize and please let me know so I can delete it. I am a programmer, and one thing we have to worry about is optimization. This is why my question probably isn’t as […]

Triangle $ABC$ and equilateral triangles $ABC'$, $BCA'$ and $ACB'$.

We consider a triangle $ABC$ whose angles are less then $120°$ and construct the equilateral triangles $ABC’$, $BCA’$ and $ACB’$, exterior to $ABC$. $I$ denotes the intersection of $(AA’)$ and $(CC’)$. 1) Show that $AA’=BB’=CC’$. 2) Show that $\widehat{BIC}=\widehat{BIA}=120°$. 3) Show that the lines $(AA’)$, $(BB’)$ and $(CC’)$ intersect. I have no idea how to […]