## Morphism of distinguished triangles where one of the arrows is a quasi-isomorphism

Let R be any ring and let A→B→C→[1] and A′→B′→C′→[1] be distinguished triangles of complexes of R-modules. Let f:A→A′, g:B→B′ and h:C→C′ be morphism of complexes such that (f,g,h) is a morphism of distinguished triangles. I wonder whether the following statement is true: If f:A→A′ is a quasi-isomorphism, then there is a quasi-isomorphism cone(g)∼→cone(h). Clearly, … Read more

## The outer Nagel points and unknown central circle

Na, Nb, Nc are the outer Nagel points. A’B’C’ is the contact triangle. I claim that lines A’B’, A’C’, B’C’ always cut the sides of the triangle NaNbNc at six points corresponding to an unknown circle. Surprisingly its center is not defined in the ETC: Is it safe to say that this is actually the … Read more

## Routh’s theorem in three dimensions

The most well known case of Routh’s triangle theorem is: If the sides BC, CA,and AB are trisected at the points D, E, and F, respectively, then the area of the inside triangle formed by AD, BE, CF is $\dfrac{1}{7}$th of the area of that of the triangle ABC. Here is my question: can Routh’s … Read more

## Point of concurrency [closed]

Closed. This question is off-topic. It is not currently accepting answers. Want to improve this question? Update the question so it’s on-topic for MathOverflow. Closed last year. Improve this question I am looking for the proof of the following claim: Claim: Let $\triangle ABC$ be an arbitrary triangle, $D$ its nine-point center and $E,F,G$ are … Read more

## Vertices of hyperbolic triangle with given angles

This is probably a well-known problem in hyperbolic geometry, but here goes anyway. In the Poincar’e upper-half plane model, I am given three angles α, β, and γ with α+β+γ<π. I would like to find vertices A, B, and C of a hyperbolic triangle so that the respective angles are α, β, and γ. Of … Read more

## About the ‘minimum triangle’ which includes a convex bounded closed set

Question : Is the following true? “Letting K be a convex bounded closed set on a plane, then there exists a triangle M, which includes K, such that |M|≤2|K|. Here, |M|,|K| is the area of M,K respectively.” Motivation : First, I’ve thought about the case that K is a parallelogram. Then, I reached the above … Read more

## Distance between point inside a triangle and its vertices [closed]

Closed. This question is off-topic. It is not currently accepting answers. Want to improve this question? Update the question so it’s on-topic for MathOverflow. Closed 2 years ago. Improve this question How to determine the distance between an arbitrary point inside a triangle and its vertices if side lengths are given. Is there any correlation … Read more

## Triangle angle bisectors, trisectors, quadrisectors, …

With the triangle angle bisector theorem and Morley’s trisector theorem as background, are there any pretty theorems known for triangle n-sectors, n > 3? For example, angle quadrisectors? The images below suggest a theorem which I’m hesitant to believe, but illustrates what I seek: Q1. Do the \tfrac{1}{4} rays (brown) meet the \tfrac{1}{2} rays (red) … Read more

## How often can subsets of a universe intersect exactly once?

My question is inspired by the following observation: Claim: It is not possible to choose n subsets of the universe [n], each of size Ω(n), such that for each subset S and each element s∈S, there is another subset S′ such that S∩S′={s}. Proof: Suppose, towards a contradiction, that this system of subsets exists. First, … Read more

## Malfatti Circles – Limiting point

“Three circles packed inside a triangle such that each is tangent to the other two and to two sides of the triangle are known as Malfatti circles” (for a brief historical account on this topic, see here and here on MathWorld). Consider the triangle formed by the centers of these circles, one can draw a … Read more