It is well known that the medians of a triangle divide it into 6 triangles.
It is less known that the centers of their circumscribed circles belong to one circumference as drawn below

This remarkable theorem  was proved in the 21st century! Unfortunately, I lost its source.
I can't prove this difficult  theorem by hand. However, I can prove it with Maple.
The aim of this post is to expose these proofs. Everybody knows that it is scarcely possible
to construct a general triangle with help of the geometry package of Maple.
Without loss of generality one may assume that the vertex A is placed at the origin,
the vertex B is placed at (1,0), and the vertex C(xC,yC). We firstly consider the theorem
in the case of concrete values of xC and yC.

restart; with(geometry):with(plots):
point(A, 0, 0);
point(B, 1, 0);
xC := 15*(1/10); yC := sqrt(3); point(C, xC, yC);
triangle(T, [A, B, C]);
median(mA, A, T, MA);
median(mB, B, T, MB);
median(mC, C, T, MC);
line(m1, [A, MA]);
line(m2, [B, MB]);
intersection(E, m1, m2);
triangle(AEMB, [A, E, MB]);
circumcircle(c1, AEMB, 'centername' = C1);
circumcircle(c2, triangle(CEMB, [C, E, MB]), 'centername' = C2);
circumcircle(c3, triangle(CEMA, [C, E, MA]), 'centername' = C3);
circumcircle(c4, triangle(BEMA, [B, E, MA]), 'centername' = C4);
circumcircle(c5, triangle(BEMC, [B, E, MC]), 'centername' = C5);
circumcircle(c6, triangle(AEMC, [A, E, MC]), 'centername' = C6);
circle(CC, [C1, C2, C3]);
IsOnCircle(C4, CC);
                              true

IsOnCircle(C5, CC);
                              true
IsOnCircle(C6, CC);
                              true
display([draw([T(color = black), mA(color = black), mB(color = black), mC(color = black), C1(color = blue), C2(color = blue), C3(color = blue), C4(color = blue), C5(color = blue), C6(color = blue), CC(color = red)], symbol = solidcircle, symbolsize = 15, thickness = 2, scaling = constrained), textplot({[-0.5e-1, 0.5e-1, "A"], [.95, 0.5e-1, "B"], [xC-0.5e-1, yC+0.5e-1, "C"]})], axes = frame, view = [-.1 .. max(1, xC)+.1, 0 .. yC+.1]);

This can be done as a procedure in such a way.

restart; SixPoints := proc (xC, yC) geometry:-point(A, 0, 0); geometry:-point(B, 1, 0); geometry:-point(C, xC, yC); geometry:-triangle(T, [A, B, C]); geometry:-median(mA, A, T, MA); geometry:-median(mB, B, T, MB); geometry:-median(mC, C, T, MC); geometry:-line(m1, [A, MA]); geometry:-line(m2, [B, MB]); geometry:-intersection(E, m1, m2); geometry:-triangle(AEMB, [A, E, MB]); geometry:-circumcircle(c1, AEMB, 'centername' = C1); geometry:-circumcircle(c2, geometry:-triangle(CEMB, [C, E, MB]), 'centername' = C2); geometry:-circumcircle(c3, geometry:-triangle(CEMA, [C, E, MA]), 'centername' = C3); geometry:-circumcircle(c4, geometry:-triangle(BEMA, [B, E, MA]), 'centername' = C4); geometry:-circumcircle(c5, geometry:-triangle(BEMC, [B, E, MC]), 'centername' = C5); geometry:-circumcircle(c6, geometry:-triangle(AEMC, [A, E, MC]), 'centername' = C6); geometry:-circle(CC, [C1, C2, C3]); return geometry:-IsOnCircle(C4, CC), geometry:-IsOnCircle(C5, CC), geometry:-IsOnCircle(C6, CC), geometry:-draw([CC(color = blue), C1(color = red), C2(color = red), C3(color = red), C4(color = red), C5(color = red), C6(color = red), T(color = black), mA(color = black), mB(color = black), mC(color = black), c1(color = green), c4(color = green), c2(color = green), c3(color = green), c5(color = green), c6(color = green)], symbol = solidcircle, symbolsize = 15, thickness = 2) end proc;
SixPoints(1.5, 1.2);

true, true, true, PLOT(...)
 SixPoints(1.5, 1.2)[4];

See geom1.mw

To be continued (The general case will be considered in  part 2http://www.mapleprimes.com/posts/200210-Six-Points-On-Circumference-2 .).

computer a Gröbner basis for <f_[1] = x^2*y - 2*y*z + 1, f_[2] = x*y^2-z^2+ 2*x,  f_[3] = y^2*z - x^2+ 5 > belong to Q[x,y,z], using ≺= <_grlex with x≺y≺z. compare your output to the Gröbner basis the Maple computers with a different order.

I have no idea where to start with this problem.

I know how to create arrays, but I want to assign characteristics to each position in the array; the array being a population with each member having a set of characteistics like height, weight etc. I want to be able to have some of the characteristics able to change with time, and some remain constant (those that change with time will be defined by generic equations)

I have tried looking on maplesoft for answers, and the help within maple 16, but have been unable to make any progress.

Sorry if the question is not specific enough.

Thanks in advance x

Please I need programme to solve this nonlinear couple ode with values of k_1 from 1.5 to 2.5 and eta from 0.0 to 2.0 real_work_3.mw Thanks for your help.

Hi all,

We're programming with Maple objects, and would like to have a Maple object that is appliable (i.e. acts as an operator).  We're really after partial differential operator (PDO) objects, but I give a simpler example below.  The problem is this: the ?ModuleApply proc needs to be non-static (so that each object has its own ?ModuleApply), and it may want to refer to the object instance, via ?thismodule.  Code like this works great when it is typed at the command line or read in from .mpl file, but it barfs as soon as saved to a .mla.  Why is that?  And is there an easy fix / workaround?

Here is an example worksheet, that creates a basic "compute mod p" object

BugThismoduleMLA.mw

The issue seems to be that, for code typed at command line, ?thismodule is resolved when I would expect, i.e. in the instance object; but saving to the .mla seems to cause ?thismodule to be resolved then-and-there, which means it resolves to the prototype object.  For this particular case there is an easy workaround, but we really want to refer to ?thismodule if possible.  Is there a way to do that?

BTW I'm running Maple 17.02 (sorry don't have access to 18 yet).

Thanks,
Ian and Tracy

I took a calculus 1 class in 2002, so i have many maple worksheets i would like to view on my new dell venue 8 pro. can the player read the .mws ext ?  Donald Altringer

ps. I have maple 8 on my laptop but not on th tablet and i don't  have a way of installing it

I took a calculus 1 class in 2002, so i have many maple worksheets i would like to view on my new dell venue 8 pro. can the player read the .mws ext ?  Donald Altringer

ps. I have maple 8 on my laptop but not on th tablet and i don't  have a way of installing it

Dear all

Please I need your help to simplify by the coefficient a in this Matrix

I have The matrix A defined by  A:=Matrix(2,2, [[a,a],[3*a,4*a]]);

I want with maple transform A to  A:=a*Matrix(2,2, [[1,1],[3,4]]);

Thanks for your Help.

I try to solve these two nonlinear odes, but no desire outcome. Please helpreal_work1.mwreal_work1.mw  or real_work1.mw

Dear all,

Thanks for your answer. I have a simple question:

Let A be a Matrix, X[1] and X[2] two vectors.

I have this equation:  X[2]= X[1]+ A*X[1]+A*X[2];  Using Maple how can I  writte X[2] =P*X[1]; where P is a matrix to be founded.

Here, P:=(Id-A)^(-1)*A; But how using maple.

I have tried to use Maple to solve ordinary differential equations but i have this error. Could you please help me to fix this problem

> eq1 := diff(v1(t), t) = v2(t);
> eq2 := diff(v2(t), t) = -v1(t)+(3*(v1(t)^2-1))*v2(t);
> init1 := v1(0) = 2;
> init2 := v2(0) = 0;
> with(DEtools);
> DEplot({eq1, eq2}, [v1(t), v2(t)], 0 .. 3*Pi, {[0, 0, 0]}, scene = [v1, v2], stepsize = .1);

Error, (in DEtools/DEplot/direction) division by zero

thank you

b := 1;

h := 1;

A := b*h;

E := 210*10^9;

qr := 100;

Dp := (1/12)*E*b*h^3;

R := 20;

teta := 3;

sys1 := {(E.A+Dp/R^2)*(diff(u(t), `$`(t, 2))) = -E*A*(diff(w(t), t))/R+Dp*(diff(w(t), `$`(t, 3)))/R, -Dp*(diff(w(t), `$`(t, 4))) = E*A*(diff(u(t), t))/R-Dp*(diff(u(t), `$`(t, 3)))/R+E*A/R-100};

dsolve(sys1);

#THİS GAVE ME THE RESULTS LİKE THAT:

{u(t) = (1/4801)*_C3*(-1+sqrt(4801))^(3/2)*exp(-(1/20)*sqrt(-1+sqrt(4801))*t)-(1/4801)*_C4*(-1+sqrt(4801))^(3/2)*exp((1/20)*sqrt(-1+sqrt(4801))*t)-(1/4801*I)*_C5*(1+sqrt(4801))^(3/2)*exp(-(1/20*I)*sqrt(1+sqrt(4801))*t)-(1/4801)*_C6*(-1+sqrt(4801))^(3/2)*exp((1/20)*sqrt(-1+sqrt(4801))*t)+(1/4801)*_C3*sqrt(-1+sqrt(4801))*exp(-(1/20)*sqrt(-1+sqrt(4801))*t)-(1/4801)*_C4*sqrt(-1+sqrt(4801))*exp((1/20)*sqrt(-1+sqrt(4801))*t)+(1/4801*I)*_C5*sqrt(1+sqrt(4801))*exp(-(1/20*I)*sqrt(1+sqrt(4801))*t)-(1/4801)*_C6*sqrt(-1+sqrt(4801))*exp((1/20)*sqrt(-1+sqrt(4801))*t)-(104999999/105000000)*t+_C1, w(t) = _C2+_C3*exp(-(1/20)*sqrt(-1+sqrt(4801))*t)+_C4*exp((1/20)*sqrt(-1+sqrt(4801))*t)+_C5*exp(-(1/20*I)*sqrt(1+sqrt(4801))*t)+_C6*exp((1/20)*sqrt(-1+sqrt(4801))*t)}

#SO THERE ARE 6 UNKOWN(C1...C6) SO I NEED 6 CONDITION. MY BOUNDRY CONDITIONS:

bs := u(0) = 0, (D(u))(0) = 0, w(0) = 0, (D(w))(0) = 0, w(teta) = 0, (D(w))(teta) = 0;

> r := dsolve({bs, sys1});
%;
Error, (in dsolve) invalid arguments; expected an equation, or a set or list of them, received: {{-17500000000*(diff(diff(diff(diff(w(t), t), t), t), t)) = 10500000000*(diff(u(t), t))-875000000*(diff(diff(diff(u(t), t), t), t))+10499999900, 210043750000*(diff(diff(u(t), t), t)) = -10500000000*(diff(w(t), t))+875000000*(diff(diff(diff(w(t), t), t), t))}}
>

I tried to change the equations row. But ı dont understand where something wrong.I tried so many times. May u please find where ıt is wrong? 

Hello,

I would like to know whether it is possible to change the color of animations that have already been produced and assigned to a variable.

Let's take for example

P := plots:-animate(plot, [sin(x+t), x = -Pi .. Pi], t = -Pi .. Pi, frames = 8)

is there a command to change the color of the curve AFTER it has been produced?

The calculus to generate all my plots took 4 hours so I would like not to have them calculated again.

Thx

How can I typeset this in Maple? A ket with a 2 element column vector in it, but without the vector brakets. Like this:

| x over y >

Thanks

AOA... Pl correct it

Help.mw