OpenLLM-France/wikimedia huggingface dataset:
# Exactly solvable 2d CFTs
## Spectrums and loop CFTs

See Sections 2.1 and 2.4 of Ref.

### Course questions

* In the Ising minimal model, what are the OPE spectrums? Write the channel spectrums of all inequivalent 4-point functions.
* In a generalized minimal model, write the 3 channel spectrums of $\left\langle V_{\langle 3,3\rangle }^{d}V_{\langle 2,4\rangle }^{d}V_{\langle 1,4\rangle }^{d}V_{\langle 2,5\rangle }^{d}\right\rangle$.
* What are the largest possible non-diagonal spectrums for a CFT with 2 degenerate fields, such that all spins are integer?

### ESDI: Conformal dimensions of non-diagonal fields

We consider the non-diagonal primary fields with integer conformal spins, in a CFT that has a degenerate field $V_{\langle 1,3\rangle }^{d}$. We assume $\Re \beta ^{2}>0$.
* Which non-diagonal fields $V_{(r,s)}$ can exist?
* Show that the set of their total conformal dimensions $\Delta +{\bar {\Delta }}$ is discrete and bounded from below.
* Show that there are finitely many fields whose total dimension is below any given bound.
* Assuming $\beta ^{2}\in \mathbb {R}$, sketch the set of non-diagonal fields on a plot whose axes are $\Delta -{\bar {\Delta }}$ and $\Delta +{\bar {\Delta }}$.

## Liouville theory and non-diagonal minimal models
### Liouville theory

See Section 2.2.2 of Ref.

### D-series minimal models

See Section 2.3.1 of Ref.

### Course questions

* From OPEs in generalized minimal models, deduce the OPEs in Liouville theory with $c\leq 1$. In particular, show that the measure of integration over momentums is the uniform measure $dP$.
* What is the intersection of the spectrums of the A-series and D-series minimal models at the same central charge? Is this the spectrum of a consistent CFT?
* Show that the spectrum of the D-series minimal model DMM$_{6,5}$ is a subset of the spectrum of the 3-state Potts CFT. Explain why the diagonal primary field $V_{P_{(0,{\frac {1}{2}})}}$ appears in the non-diagonal sector of DMM$_{6,5}$. Admitting that $V_{P_{(0,{\frac {1}{2}})}}$ transforms in the standard representation of the symmetric group $S_{3}$, explain why it appears twice in DMM$_{6,5}$.

### ESES: E-series minimal models

* Find the spectrums of E-series minimal models in Wikipedia.
* Which diagonal degenerate fields exist in E-series minimal models?
* Compare the spectrums of the A-series, D-series and E-series minimal models with the same central charge.
* In the case $q=12$, rewrite the spectrum in terms of non-diagonal fields $V_{(r,s)}$, and compare with loop models.
# History of Sri Lanka/Mahinda Rajapaksa

Mahinda Rajapaska (born 18 November 1945) is the 6th President of Sri Lanka and Commander in Chief of the Sri Lankan Armed Forces. A lawyer by profession, Rajapaksa was first elected to the Parliament of Sri Lanka in 1970, and served as Prime Minister from 6 April 2004 until his victory in the 2005 Presidential election. He was sworn in for a six-year term as president on 19 November 2005. He was re-elected for a second term in office on 27 January 2010.
# History of Sri Lanka

Sri Lanka, officially known as the Democratic Socialist Republic of Sri Lanka, is an island nation located in South Asia, in the Indian Ocean, southeast of the Indian subcontinent. The country has a rich history dating back over 3,000 years, marked by its ancient kingdoms, diverse cultures, and significant contributions to art, literature, and religion. It is known for its stunning landscapes, including tropical forests, mountains, and beaches. Sri Lanka is famous for its tea plantations, wildlife, and historical sites such as the ancient city of Anuradhapura and the rock fortress of Sigiriya. The population of around 22 million is primarily Sinhalese, with Tamil and Muslim minorities. Buddhism is the predominant religion, and the official languages are Sinhala and Tamil. Sri Lanka gained independence from British rule in 1948 and became a republic in 1972.

## Lists

* History of Sri Lanka/List of presidents

## Subpages

Special:Prefixindex/History of Sri Lanka/

## See Also

* History of Asia
# Triangle geometry/Pythagorean theorem/Fact

In a right triangle,
hypotenuse
# Triangle geometry/Pythagorean theorem/Fact/Proof

We denote the vertices of the triangle by $A,B,C$, with the right angle located at $C$. We set $v=C-A$, and $w=B-C$. The vector from $A$ to $B$ equals $v+w$, and $v$ and $w$ are perpendicular to each other. Therefore,
    $\Vert {v+w}\Vert ^{2}=\left\langle v+w,v+w\right\rangle =\left\langle v,v\right\rangle +2\left\langle v,w\right\rangle +\left\langle w,w\right\rangle =\Vert {v}\Vert ^{2}+\Vert {w}\Vert ^{2}\,.$
# Cathetus theorem/Linear algebra/Fact

Let $A,B,C$ be a right triangle, with the right angle located at $C$. Let $h$ denote the altitude through $C$, and let $D$ denote the foot of this altitude on the line given by $A$ and $B$.
    $d(A,C)^{2}=d(A,D)\cdot d(A,B)\,.$
# Cathetus theorem/Linear algebra/Fact/Proof

We set $v=C-A$, and $w=B-C$. The vector from $A$ to $B$ equals $v+w$. We write the foot of the altitude as
    $D=A+\lambda (v+w)\,,$
with $\lambda \in \mathbb {R}$; the vector of the altitude has the form
    $h=C-D=A+v-A-\lambda (v+w)=v-\lambda (v+w)\,.$
The orthogonality condition for the altitude yields
    ${\begin{aligned}0&=\left\langle v-\lambda (v+w),v+w\right\rangle \\&={\left(1-\lambda \right)}\left\langle v,v\right\rangle -\lambda \left\langle w,w\right\rangle \\&=\left\langle v,v\right\rangle -\lambda {\left(\left\langle v,v\right\rangle +\left\langle w,w\right\rangle \right)};\end{aligned}}$
therefore,
    $\lambda ={\frac {\left\langle v,v\right\rangle }{\left\langle v,v\right\rangle +\left\langle w,w\right\rangle }}\,.$
Hence, we obtain
    ${\begin{aligned}d(A,D)\cdot d(A,B)&=\lambda \Vert {v+w}\Vert \cdot \Vert {v+w}\Vert \\&=\lambda \left\langle v+w,v+w\right\rangle \\&=\lambda {\left(\left\langle v,v\right\rangle +\left\langle w,w\right\rangle \right)}\\&=\left\langle v,v\right\rangle \\&=\Vert {v}\Vert ^{2}\\&=d(A,C)^{2}.\end{aligned}}$
# Triangle geometry/Law of cosines/Fact

Let a triangle $(A,B,C)$ be given, with side lengths $a,b,c$. Let $\gamma$ denote the angle at $C$.
    $c^{2}=a^{2}+b^{2}-2ab\cos \gamma \,$
# Intercept theorem/Two rays/Fact/Proof

Without loss of generality, we may assume that $s_{1}=A_{1}$, $s_{2}=A_{2}$, and $v=A_{2}-A_{1}$, as this does not change the lines involved. We write $B_{1}=tA_{1}$. We have $A_{2}=A_{1}+v$; therefore, we obtain
    $tA_{2}=tA_{1}+tv=B_{1}+tv\,.$
This point belongs to $S_{2}$ and also to $H$. This means that this point is just $B_{2}$. Hence, $B_{2}=tA_{2}$, and
    ${\frac {\Vert {B_{1}-B_{2}}\Vert }{\Vert {B_{1}}\Vert }}={\frac {\Vert {tA_{1}-tA_{2}}\Vert }{\Vert {tA_{1}}\Vert }}={\frac {\Vert {A_{1}-A_{2}}\Vert }{\Vert {A_{1}}\Vert }}\,.$
# Intercept theorem/Two rays/Similarity of triangles/Fact

Let $V$ denote a two-dimensional Euclidean vector space, and let $s_{1},s_{2},v\in V$ be vectors and suppose that $v$ and $s_{1}$ are linearly independent, and that $v$ and $s_{2}$ are linearly independent. Let $S_{1}=\mathbb {R} s_{1}$ and $S_{2}=\mathbb {R} s_{2}$ denote the lines defined by $s_{1}$ and $s_{2}$. Let $P$ and $Q$ be points in $V$ with the corresponding parallel lines $G=P+\mathbb {R} v$ and $H=Q+\mathbb {R} v$. We denote the intersecting points (which exist uniquely due to the condition) by
    $A_{1}=S_{1}\cap G,\,A_{2}=S_{2}\cap G,\,B_{1}=S_{1}\cap H,\,B_{2}=S_{2}\cap H,$
and suppose that $A_{1},A_{2},A_{1}-A_{2}\neq 0$.
    ${\frac {d(B_{1},B_{2})}{d(A_{1},A_{2})}}={\frac {\Vert {B_{1}}\Vert }{\Vert {A_{1}}\Vert }}={\frac {\Vert {B_{2}}\Vert }{\Vert {A_{2}}\Vert }}\,.$
# Intercept theorem/Two rays/Fact

Let $V$ denote a two-dimensional Euclidean vector space, and let $s_{1},s_{2},v\in V$ be vectors and suppose that $v$ and $s_{1}$ are linearly independent, and that $v$ and $s_{2}$ are linearly independent. Let $S_{1}=\mathbb {R} s_{1}$ and $S_{2}=\mathbb {R} s_{2}$ denote the lines (the rays) defined by $s_{1}$ and $s_{2}$. Let $P$ and $Q$ be points in $V$ with the corresponding parallel lines $G=P+\mathbb {R} v$ and $H=Q+\mathbb {R} v$. We denote the intersecting points (which exist uniquely due to the condition) by
    $A_{1}=S_{1}\cap G,\,A_{2}=S_{2}\cap G,\,B_{1}=S_{1}\cap H,\,B_{2}=S_{2}\cap H,$
and suppose that $A_{1},B_{1}\neq 0$.
    ${\frac {d(B_{1},B_{2})}{\Vert {B_{1}}\Vert }}={\frac {d(A_{1},A_{2})}{\Vert {A_{1}}\Vert }}\,.$
# Intercept theorem/Two rays/Similarity of triangles/Fact/Proof

This follows directly from fact.
# Intercept theorem/Three rays/Fact

Let $V$ be a two-dimensional Euclidean vector space, let $s_{1},s_{2},s_{3}\in V$ be vectors, and suppose that $v\in V$ is linearly independent to each of these vectors. Let $S_{i}=\mathbb {R} s_{i}$, $i=1,2,3$, be the lines defined by the $s_{i}$. Let $P$ and $Q$ be points in $V$ with den corresponding parallel lines $G=P+\mathbb {R} v$ and $H=Q+\mathbb {R} v$. We denote the intersecting points of the lines (which exist uniquely due to the conditions) by
    $A_{1}=S_{1}\cap G,\,A_{2}=S_{2}\cap G,\,A_{3}=S_{3}\cap G,\,B_{1}=S_{1}\cap H,\,B_{2}=S_{2}\cap H,\,B_{3}=S_{3}\cap H,$
and suppose that $B_{1}\neq B_{2}$.
    ${\frac {d(B_{2},B_{3})}{d(A_{2},A_{3})}}={\frac {d(B_{1},B_{2})}{d(A_{1},A_{2})}}\,.$
# Intercept theorem/Three rays/Fact/Proof

We apply fact twice, to the situation given by $s_{1},s_{2}$ and to the situation given by $s_{2},s_{3}$, and we obtain
    ${\frac {d(B_{1},B_{2})}{d(A_{1},A_{2})}}={\frac {\Vert {B_{2}}\Vert }{\Vert {A_{2}}\Vert }}={\frac {d(B_{2},B_{3})}{d(A_{2},A_{3})}}\,.$
# Thales's theorem/Linear algebra/Fact

Let $P$ be a point in the Euclidean plane $E$, let $K$ be a circle with center $P$, and let $G$ be a line through $P$, intersecting the circle in the points $A$ and $B$.
$C\in K$,
$A,B,C$
$C$
# Thales's theorem/Linear algebra/Fact/Proof

Without loss of generality, we may assume that $E=\mathbb {R} ^{2}$ and $P=0$, we denote the radius by $r$. We write $w=C$, $v=A$, $B=-v$. The vector from $A$ to $C$ is $-v+w$, and the vector from $B$ to $C$ is $v+w$. Therefore,
    ${\begin{aligned}\left\langle -v+w,v+w\right\rangle &=\left\langle -v,v\right\rangle +\left\langle -v,w\right\rangle +\left\langle w,v\right\rangle +\left\langle w,w\right\rangle \\&=-\left\langle v,v\right\rangle -\left\langle v,w\right\rangle +\left\langle v,w\right\rangle +\left\langle w,w\right\rangle \\&=-\left\langle v,v\right\rangle +\left\langle w,w\right\rangle \\&=-r^{2}+r^{2}\\&=0.\end{aligned}}$
This means that the two sides of the triangle meeting at $C$ are perpendicular to each other.
# Linear algebra (Osnabrück 2024-2025)/Intercept theorem/Two rays/Similarity of triangles/Fact/Factreferencenumber

Corollary 36.17
# Linear algebra (Osnabrück 2024-2025)/Intercept theorem/Two rays/Fact/Factreferencenumber

Theorem 36.16
# MDLD/Euclidean vector space

Start=Euclidean vector space Target=Euclidean vector space/Definition
# MDLD/Euclidean plane

Start=Euclidean plane See=Euclidean vector space MDLD/Euclidean vector space Target=Euclidean vector space/Definition
# MDLD/congruent (triangle)

Start=congruent (triangle) Target=Triangle/Congruent/Definition
# Should mentally ill people be allowed to have children?

Depending on the severity of the case, people with mental illnesses are mostly incapable of taking care of others, especially children. In the case of neglectful parenting, the State´ social services intervene and regulates the tenency of the children. Should this public interference apply also to people diagnosed with a certain type of mental illness? How to establish objective parameters in order not to incur in injustices?

## Mentally ill people should be allowed to have children

Relevant details, definitions and assumptions regarding the first possibility.

### Pro

* Pro Every human being, regardless of their condition, should be able to choose their path in life, including having or not having children.
** Objection Having children is not people choosing their path in life, it involves giving rise to a new human life. Thus, it is their own path in life plus starting a new life which in turn can give rise to a branch of further humans. Furthermore, it also the path of the partner who may otherwise not have children (possibly adopting children) or have them with somebody else.
** Objection If a person is unable to take care of children they should not be allowed to have children.
** Objection Mentally ill people might do all forms of abuse on the children.
*** Objection That may apply only to some mental illnesses.
*** Objection Then it's a question if they are able to take care of their children. Being mentally ill does not negate their right to have the child--it's another whole debate if they are able to care for their children or not.
** Objection Not everybody is capable of making informed choices.
* Pro The rights of being a person entails the existence of a person. Since the primordial instinct of humans is to reproduce, every human being has the right to have children.
** Objection It's absurd to argue humans have rights if it's their primordial instinct. It's a non sequitur and does not make any sense.
** Objection Many humans do not or not anymore or not in some period like contemporary times feel a primordial instinct to reproduce.
* Pro For many mental illnesses as well as low severity of additional ones, people are well or sufficiently capable of taking care of their children.

### Con

* Con Sick people —especially mentally ill ones— have no right to force their illness on someone else by having children.
** Objection Not all mental illnesses are inherited.
* Con The natural flows of natural selection are meant to prevent the sick from having children, as they are not as robust as the healthy to reproduce and thrive.
** Objection Society has changed from the old times. Now, as we have more technologies to adapt to the objectively sick, they have the ability to survive and the right to do anything a healthy person would do.
# MDLD/similar (triangle)

Start=similar (triangle) Target=Triangle/Similar/Translation and angle-preserving/Definition
# MDLD/right triangle

Start=right triangle Target=Right triangle/Definition
# Triangle/Altitude/Definition

For a triangle $A,B,C$ in a Euclidean plane, the line running through $A$ and being perpendicular to the line given by $B$ and $C$, is called the altitude line through $A$. The line segment from $A$ to the line given by $B$ and $C$ is called the altitude through $A$.
# Triangle/Foot/Definition

In a triangle $A,B,C$ in a Euclidean plane, the intersecting point of the altitude line through $A$ with the line given by $B$ and $C$ is called the foot of this altitude.
# MDLD/altitude line

Start=altitude line Target=Triangle/Altitude/Definition
# MDLD/altitude (triangle)

Start=altitude (triangle) Target=Triangle/Altitude/Definition
# MDLD/foot (altitude)

Start=foot (altitude) Target=Triangle/Foot/Definition
# MDLD/foot of the altitude

Start=foot of the altitude Target=Triangle/Foot/Definition
# Right triangle altitude theorem/Linear algebra/Fact

Let $A,B,C$ be a right triangle, with the right angle located at $C$. Let $h$ be the altitude through $C$, and let $D$ denote the foot of this altitude through the linen given by $A$ and $B$.
    $d(C,D)^{2}=d(A,D)\cdot d(B,D)\,.$
# MDLD/hypotenuse

Start=hypotenuse Target=Right triangle/Hypotenuse/Definition
# Linear algebra (Osnabrück 2024-2025)/Part II/Lecture 37

Beside the three vertices of a triangle, there are further characteristic points of a triangle, as its centroid (center of gravity), the circumcenter, den Incenter and the orthocenter.
    Medians and centroid
Triangle geometry/Median/Introduction/Section
    Perpendicular bisectors and circumcenter
Triangle geometry/Perpendicular bisector/Introduction/Section
    Angle bisectors and incenter
Triangle geometry/Angle bisectors/Introduction/Section
    The orthocenter

## Lemma
### Proof

$\Box$

## Definition

Triangle/Orthocenter/Definition
    The Euler line
Triangle geometry/Euler line/Introduction/Section
    The nine-point circle
Triangle geometry/Nine-point circle/Introduction/Section
# Soft properties of Boolean functions

Properties of Boolean functions :
| Hard | not dependent on arity |
| Soft | dependent on arity |
ShortcutBoolf-soft :
The number of Boolean functions with arity $n$ (short for adicity $\leq n$) is $2^{2^{n}}$ (A001146).
This article is about properties of finite truth tables of Boolean functions, that change with the arity.

## simple

* weight: number of true places
* sharpness: weight parity sharp: odd weight blunt: even weight

## subsets

* noble: truth tables that are their own twins
* symmetric: order of inputs does not matter

## equivalence classes based on similarity

see also: Extended families and clans of Boolean functions

### ultra (extension with half-complement)

Each Boolean function has two half-complements. (The truth table is complemented on the left or on the right side.) A super-family has a unique half-complement. Merging them gives an ultra-family. As super-clans consist of super-families, they can be extended in the same way. (Factions do not have a unique half-complement, so it does not seem useful to define ultra-factions.)

#### ultra-family (negation, complement, half-complement)

The functions in an ultra-family have symmetric positions in a hypercube graph (or the related matrix). See matrices. This family is a complete ultra-family: 1100 1010 (better seen in its matrix)

#### ultra-clan (negation, permutation, complement, half-complement)

An ultra-clan is the merge of a super-clan and its half-complement. It can also be seen as a merge of ultra-families, that are permutations of each other. See here for a table of the 39 ultra-families of 4-ary Boolean functions.

## partitions into blocks of equal size
### consul (binary Walsh spectrum)

The Walsh spectrum of a TT is its product with a Walsh matrix. The binary Walsh spectrum of a TT is its product with a binary Walsh matrix, using F$_2$ operations (mod 2). It is always a Walsh function, and shall be called consul. The term is also used for the integer denoting the Walsh function.
The consul integer is the Walsh index of the twin prefect (prefect of the twin).
| sharp family with consuls 0...7 | blunt family with consul 2 (compare 4-ary family with consul 6) |
| 3-ary families with Walsh spectra (integers) and consuls (red backgrounds on the right) ||
tribe :
3-ary_Boolean_functions#Walsh spectrum, 4-ary_Boolean_functions#wec

### patron

The patron of a truth table is the XOR of itself and its twin. It is a noble. (3-ary images)

### praetor

XOR of left and right half of the TT. (3-ary images)

### quaestor

XOR of left and reversed right half of the truth table (i.e. of the coordinates in the Hasse matrix) (3-ary images)

## principalities and dominions

    Principalities and dominions of Boolean functions
A principality is a set of n-ary truth tables whose (n+1)-ary noble equivalents form a faction. Dominions are closely related.
# Studies of Boolean functions/terminology

dimension of a segment :

## properties

see Properties of Boolean functions and of truth tables
* clan negation and permutation equivalence class partitioned into families and factions
* family negation EC
* faction permutation EC
* splinter intersection of family and faction
* tribe Blunt TT form tribes, denoted by their consul weight.
* super Prefix used for EC of BF extended by complement.
* ultra Prefix used for EC of TT extended by half-complement.
* foible The foibles are seven properties, that correspond to the vertices of a Fano plane. (see here for an overview)
** even/odd foible of a BF, equal to first digit of TT oddness also called parity
** evil/odious foible of a BF, equal to last digit of TT odiousness also called depravity BF is odious, iff Ж has odd weight
** pretty/ugly foible of a BF, XOR of odd and odious uglyness
** blunt/sharp foible of a TT, equal to parity of TT weight sharpness
** obtuse/acute foible of a TT, similar to sharpness acuteness
** polite/rude and smooth/rough foibles of a TT, similar to sharpness and acuteness rudeness and roughness
* female/male BF is male, iff its root is sharp. gender
* prefect BF ↦ linear BF
* sub-prefect BF ↦ senior linear BF (category)
* consul TT ↦ linear TT (basically twin prefect)
* praetor XOR of left and right half of TT
* quaestor XOR of left and reversed right half of TT
* principality and dominion  juniors corresponding to noble factions

### provisional

* partner/friend BF whose TT differ only in least/most significant bit. Applies also to factions.
* squad Union of a faction with its complement, its partner and its complement partner. Always four factions.
* platoon Union of a squad and its twin. May consist of one of two squads, i.e. four or eight factions.
* company The name would make sense for the union of a platoon and its friend. May consist of one or two platoons.
* cluster That name is still used for platoon.
* federation related to clusters and great dominions

## general terms

These words are sometimes used in variable names.
* recto and verso inside and outside of a set (see also split)
* powers of two and factorial
| $n$ | $2^{n}$ | $2^{2^{n}}$ | $n!$ |
| paltry | plenty | plethora | galore |
# Triangle geometry/Median/Introduction/Section
## Definition

Finite point set/Center of gravity/Definition
This is also called the centroid of the points. It just the barycentric combination of the points where every point enters with the same weight. For two points $P,Q\in E$, the center of gravity ${\frac {1}{2}}P+{\frac {1}{2}}Q$ is also called the midpoint of the two points (or of the line segment ${\overline {P,Q}}$). In the case of two real numbers, we also talk about the arithmetic mean of the numbers. For $n=1,2,3$, the centroid of the points is also the centroid of its convex hull. The centroid of the three points is also the intersection point of the medians of the triangle.

## Definition

Triangle/Median/Definition
The median through $A$ runs through the point $A$ and the midpoint of the opposite side of the triangle.

## Theorem
### Proof

$\Box$
In particular, the centroid divides every median with the ratio $2:1$, the longer part being closer to the vertex.
# Finite point set/Center of gravity/Definition

For a set of $n$ points $P_{1},\ldots ,P_{n}$ in an affine space $E$ over a real vector space $V$, the barycentric combination
    ${\frac {1}{n}}P_{1}+{\frac {1}{n}}P_{2}+\cdots +{\frac {1}{n}}P_{n}$
is called the center of gravity of these points.
# Triangle/Median/Definition

For a nondegenerate triangle $A,B,C$ in a Euclidean plane, the line
    ${\frac {B+C}{2}}+s{\left(A-{\frac {B+C}{2}}\right)},\,s\in \mathbb {R} ,$
is called the median through $A$.
# Triangle/Medians/Intersecting point/Fact

In a nondegenerate triangle in the Euclidean plane,
medians
${\frac {A+B+C}{3}}$
# Triangle/Medians/Intersecting point/Fact/Proof

We consider the condition
    ${\frac {B+C}{2}}+s{\left(A-{\frac {B+C}{2}}\right)}={\frac {A+C}{2}}+t{\left(B-{\frac {A+C}{2}}\right)}\,,$
which yields
    ${\begin{aligned}{\frac {B-A}{2}}&=t{\left(B-{\frac {A+C}{2}}\right)}-s{\left(A-{\frac {B+C}{2}}\right)}\\&={\left(-s-{\frac {t}{2}}\right)}A+{\left(t+{\frac {s}{2}}\right)}B+{\frac {s-t}{2}}C.\end{aligned}}$
We may assume $E=\mathbb {R} ^{2}$ and $A=0$. This implies, because $B$ and $C$ are linearly independent, the identity
    $s=t\,.$
Therefore,
    ${\frac {B-A}{2}}={\left(-{\frac {3}{2}}s\right)}A+{\left({\frac {3}{2}}s\right)}B\,,$
and we obtain
    $s={\frac {1}{3}}\,.$
Hence, the intersecting point is
    ${\frac {B+C}{2}}+{\frac {1}{3}}{\left(A-{\frac {B+C}{2}}\right)}={\frac {1}{3}}A+{\frac {1}{3}}B+{\frac {1}{3}}C\,.$
Due to the symmetry of the situation, this is also the intersecting point with the third median.