Articles of geometry

Geodesics (2): Is the real projective plane intended to make shortest paths unique?

From the Wikipedia article on geodesics: In Riemannian geometry geodesics are not the same as “shortest curves” between two points, though the two concepts are closely related. The difference is that geodesics are only locally the shortest distance between points […]. Going the “long way round” on a great circle between two points on a […]

Set in $\mathbb R^²$ with an axis of symmetry in every direction

Let $A\subset \mathbb R^2$ be a set that has an axis of symmetry in every direction, that is, for any $n\in S^1$, there exists a line $D$ orthogonal to $n$, such that $A$ is invariant under the (affine) reflection of axis $D$. It is easy to show that if all of these axes intersect at […]

2013 Putnam A1 Proof understanding (geometry)

Problem A1: Recall that a regular icosahedron is a convex polyhedron having 12 vertices and 20 faces; the faces are congruent equilateral triangles. On each face of a regular icosahedron is written a non-negative integer such that the sum of all $20$ integers is $39.$ Show that there are two faces that share a vertex […]

square inscribed in a right triangle

A square of maximum possible area is circumscribed by a right angle triangle ABC in such a way that one of its side just lies on the hypotenuse of the triangle. What is the area of the square? actually the answer is given as $(abc/(a^2+b^2+ab))^2$ Please provide the approach to solve the problem.

a geometrical problem

$ABC$ is a right triangle with $∠ABC=90^0$ with $AB=30 \sqrt{3}$ and $BC=30$ . $D$ is a point on segment $B$C such that $AD$ is the median. $E$ is a point on segment $AC$ such that $BE$ is perpendicular to $AC$ . $AD$ and $BE$ intersect at $F$ . what is the value of $EF$ ?

Area of Shaded Region

How would I calculate the area of the shaded region of a circle with radius 6 and length of chord AB is 6.

A line through the centroid G of $\triangle ABC$ intersects the sides at points X, Y, Z.

I am looking at the following problem from the book Geometry Revisited, by Coxeter and Greitzer. Chapter 2, Section 1, problem 8: A line through the centroid G of $\triangle ABC$ intersects the sides of the triangle at points $X, Y, Z$. Using the concept of directed line segments, prove that $1/GX + 1/GY + […]

Diophantine approximation – Closest lattice point to a line (2d)

Consider a 2D line $A x + B y + C = 0$ with integer coefficients $A, B, C$. Find the lattice point $(x, y)$ closest to the line, such that $|x|, |y| \leq n$ for some integer $n$. ($x$ and $y$ are integers, of course). It is given that the line intersects the $y$ […]

Line joining the orthocenter to the circumcenter of a triangle ABC is inclined to BC at an angle $\tan^{-1}(\frac{3-\tan B\tan C}{\tan B-\tan C})$

Show that the line joining the orthocenter to the circumscribed center of a triangle ABC is inclined to BC at an angle $\tan^{-1}\left(\frac{3-\tan B\tan C}{\tan B-\tan C}\right)$ I let the foot of perpendicular from A,B,C to opposite sides is D,E,F.Then $$\tan B=\frac{AD}{BD},\tan C=\frac{AD}{CD}$$ $$\frac{3-\tan B\tan C}{\tan B-\tan C}=\frac{3-\frac{AD}{BD}\frac{AD}{CD}}{\frac{AD}{BD}-\frac{AD}{CD}}$$ I think this way i cannot get […]

Why are Euclid axioms of geometry considered 'not sound'?

The five postulates (axioms) are: “To draw a straight line from any point to any point.” “To produce [extend] a finite straight line continuously in a straight line.” “To describe a circle with any centre and distance [radius].” “That all right angles are equal to one another.” “That, if a straight line falling on two […]