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18 votes
1 answer
644 views

Egalitarian measures

A question I got asked I while ago: If $T$ is a triangle in $\mathbb R^2$, is there a function $f:T\to\mathbb R$ such that the integral of $f$ over each straight segment connecting two points in the ...
Mariano Suárez-Álvarez's user avatar
9 votes
1 answer
297 views

Equational theory of the orthocenter

Previously asked at MSE: Briefly speaking, I'm looking for a description of the equational theory of the orthocenter function, $\mathsf{orth}$. By $\mathsf{orth}$ I mean the (partial) function sending ...
Noah Schweber's user avatar
6 votes
0 answers
121 views

How many equilaterals have vertices intersections of angle trisectors of a triangle?

The celebrated Morley’s theorem ensures that the interior trisectors, proximal to sides respectively, meet at vertices of an equilateral. In the paper Trisectors like Bisectors with Equilaterals ...
Spiridon Kuruklis's user avatar
4 votes
0 answers
144 views

Is the orthocenter "(roughly) equationally finitely-based"?

Let $T$ be the "almost everywhere" equational theory of the orthocenter function, "tweaked appropriately" to avoid partiality issues (see this earlier question of mine for details)....
Noah Schweber's user avatar
2 votes
0 answers
84 views

Another variant of the Malfatti problem

We try to add to A Variant of the Malfatti Problem As stated in the Wikipedia entry on Malfatti circles, it is an open problem to decide, given a number $n$ and any triangle, whether a greedy method ...
Nandakumar R's user avatar
  • 5,979
35 votes
17 answers
6k views

Which theorems have Pythagoras' Theorem as a special case?

Loomis famously wrote hundreds of proofs of Pythagoras' Theorem (reference below), but these are all basically proofs "from below". Today on Twitter @panlepan mentioned Carnot's theorem ...
25 votes
6 answers
2k views

Are there infinitely many "generalized triangle vertices"?

Briefly, I'd like to know whether there are infinitely many "generalized triangle centers" which - like the orthocenter - are indistinguishable from a vertex of the original triangle. This ...
Noah Schweber's user avatar
10 votes
2 answers
764 views

Generalization of Stewart's theorem?

I'm curious about the generalization of Stewart's theorem to more dimensions. MathWorld mentions that there is a generalization done by Bottema, but I could not find much information on it. All I ...
Tom D.'s user avatar
  • 163
6 votes
2 answers
215 views

Partition of polygons into 'strongly acute' and 'strongly obtuse' triangles

Definition: Let us refer to obtuse triangles with the largest angle strictly above a given cutoff value as 'strongly obtuse' - the definition is parametrized by the cutoff value. Likewise, strongly ...
Nandakumar R's user avatar
  • 5,979
5 votes
1 answer
782 views

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 ...
Wiley's user avatar
  • 667
4 votes
2 answers
575 views

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, ...
Mark B Villarino's user avatar
4 votes
2 answers
320 views

Inequality from a point in plane to a triangle OR Inequality on a quadrilateral

If points $A$, $B$, $C$ form a triangle in euclidean space and $D$ is another point in the plane of the triangle, the problem is to show that : $\frac{AB}{DA + DB} + \frac{BC}{DB + DC} \ge \frac{AC}{...
Ritesh Ahuja's user avatar
3 votes
1 answer
152 views

Triangles that can be cut into mutually congruent and non-convex polygons

It is easy to note that an equilateral triangle can be cut into 3 mutually congruent and non-convex polygons (replace the 3 lines meeting at centroid and separating out the 3 congruent quadrilaterals ...
Nandakumar R's user avatar
  • 5,979
3 votes
0 answers
905 views

A generalization of the Sawayama-Thebault theorem

1. Introduction The Sawayama-Thebault theorem is one of the best nice theorem in plane geometry. The theorem has a long history. It was published in AMM in 1938 the first solution appeared in 1973 ...
Đào Thanh Oai's user avatar
3 votes
1 answer
507 views

An new equilateral triangle related to the Morley triangle

Morley equilateral triangle is the nice theorem in Eulidean Geometry. I found an equilateral triangle and a group circle related to the Morley triangle and angle trisectors: Let $ABC$ be a triangle ...
Đào Thanh Oai's user avatar
1 vote
0 answers
202 views

Some Problems On Apollonian Gasket

Since 2013, I found Some problems on Apollonian Gasket as following. These problem also is higher level of Eppstein Point. I am looking for a proof of one of these problems: Let three $(A)$, $(B)$, $(...
Đào Thanh Oai's user avatar
1 vote
0 answers
96 views

Pseudo-Droz-Farny circles

I would like to present a construction of 2 circles. These 2 circles are somewhat similar in appearance to the well known Droz-Farny circles that can be drawn for every isogonal conjugate pairs of ...
A.Zakharov's user avatar