The dsolve command returns its solution in the form of an "equation," the left-hand side being the name of the dependent variable. The plot command wants just the expression on the right-hand side of that "equation."

plot(rhs(solution),t=.......)

Student:-VectorCalculus:-TangentPlane((x - 1)^2 + (y - 3)^2 + (z - 5)^2 = 13^2, Vector(L[1])) => -26*x - 312 = 0

where the list of points, called "list" in the OP, has been changed to the shorter name L.

Clearly, all members of the set of tangent planes can be obtained by a suitable sequence or loop. And, yes, this is a way to get an answer quickly, but if additional insight was the goal, then more work needs to be done, as the other responses show.

For many years I have used a third-party piece of software to manage the AEM ebook files, and the many other pieces of content I've created for Maplesoft. Consider the Abacre product "Advanced Find and Replace" at a cost of $30 US. I could not have survived nearly 20 years of such maintenance without this tool. It will look at worksheets and documents inside folders, finding any phrases one would see if such files were opened with a text editor. On occasion, I've even used the "Replace" functionality successfully.

Before finding this piece of software, I tried the Unix command "grep" but did not find that tool anywhere as useful.

The following 12 lines of Maple code verify the lemma for an arbitrary triangle. The following solution is an example of conceptualizing mathematics and letting a tool like Maple implement that thinking.

There is no loss of generality if one vertex of triangle ABC is placed at the origin, and one along the positive x-axis. Point E is the interior point in triangle ABC. The areas of triangles ACE and BCE are found by the formula "half the base times height." Hence, the lengths of the altitudes from E must be found. Point E is projected onto the edges AC and BC, respectively, and the distance from E to these projections provides those heights. The ratio of the areas of triangles ACE and BCE is readily found, and is the left-hand side of the equality to be demonstrated.

Point p3, the dividing point on edge AB, is found by intersecting the line through C and E with the line through A and B. The ratio of the distances from A to p3 and p3 to B forms the right-hand side of the equality to be demonstrated.

We divide the right-hand side (the ratio of areas) by the left-hand side, in the expectation that this will be 1. And it is, modulo the appropriate simplification. I was both amazed and delighted that Maple was able to manipulate the requisite symbolic quantities without undue difficulty. I don't know if the geometry package can implement such a symbolic solution - I didn't try it because I'm much more familiar with the Lines&Planes tools in MultivariateCalculus than I am with those in the geometry package. I'd be happy to learn the extent of the power of the geometry package.

with(Student:-MultivariateCalculus):
A,B,C,E:=[0,0],[b,0],[cx,cy],[u,v]:
p1:=Projection(E,Line(A,C)):
p2:=Projection(E,Line(B,C)):
h1:=simplify(Distance(E,p1)):
h2:=simplify(Distance(E,p2)):
A1:=Distance(A,C)*h1/2:
A2:=Distance(B,C)*h2/2:
R1:=simplify(A1/A2):
p3:=GetIntersection(Line(A,B),Line(C,E)):
R2:=simplify(Distance(A,p3)/Distance(p3,B)):
simplify(R1/R2) assuming real
 

Maple can display a "plus minus" symbol, but (as far as I know) Maple cannot calculate with it. There are two ways to get the symbol displayed:

In 1D math, use &+-. In 2D math, the same code works, but also the keyboard equivalent is to type pm and use command completion (the Escape key); select the +- symbol from the pop-up list.

But again, that symbol is display notation only. In Maple, there is no mechanism I know of that will allow computation with such notation. That's why Maple expresses solutions of a quadratric equation,, not with with the pm symbol, but rather as two separate expressions.

Type lambda^m (in typeset, that is, 2D, math); select, and convert to an Atomic Variable either from the keyboard (Control+Shift+a) or from the Format Menu (Convert To/Atomic Variable).

This converts any collection of symbols into a valid Maple name. If "Atomic Variables" is selected in the View Menu, the Atomic Variables in a worksheet will display in a reddish color (magenta?). If an Atomic Variable is lprinted, its MathML coding will be revealed.

This is a useful device. The downside is that each time the Atomic Variable is referenced or used, it must be re-created. However, copy/paste does work and can lessen the tedium of using these names.

with(Student:-MultivariateCalculus):

L:=Line([1,-2],[-2,3]):

Projection([1,1],L) => [-11/34, 7/34]

Distance([1,1],L) => 9/sqrt(34)

Just before test2, z is defined to be a vector. The equivalent is not done for x in test1. The error message might be cryptic, but makes sense.

@zenterix 

The Multivariate Calculus Study Guide does not use the Physics Vectors package where _i is given special meaning.

Since this guide was written, a great deal of new material has been added to the Physics package. The purpose of the guide is to exposit the Student MultivarivariateCalculus package, not the Physics package. It would be an interesting exploration to see the extent to which the Physics package could supplant the MultivariateCalculus package. It might work for the unit on vectors, lines, and planes, but there are many other topics in multivariate calculus that would be more difficult to tackle in a syntax-free way using the Physics package.

In the typical multivariate calculus course, mastering the syntax of the Physics package would probably be an overload on the students. The syntax of the MultivariateCalculus package is simpler, and there are far more syntax-free options available in the Context Panel for that package than for the Physics package.

https://www.maplesoft.com/Applications/

This site can be reached directly, or from the Maplesoft site.

Using Typeset math (2D Math) in a Document, type b

Shift 6 for an exponent

Click on the perpendicular symbol in the Common Symbols palette

Select and convert all to an Atomic Variable (Use keyboard or Format Menu/Convert To)

Press the Enter key to get an echo of the symbol, labeled with an equation label

lprint that output (reference via the equation label)

The result of the lprint is `#msup(mi("b"),mo("⊥"))`

plots:-display(plots:-textplot([2, 2, `#msup(mi("b"),mo("⊥"))`])) will draw a graph with the desired symbol printed upright, not slanted as if italic. Moreover, since the symbol is now an Atomic Variable, it can be assigned to  - it is now a valid Maple name.

The Conic Sections Tutor in the Student Precalculus package will provide the graph of this ellipse, and report the coordinates of the foci, vertices, center, and equation of the directrix. No coding or programming needed.

The following handfull of Maple commands will provide the tangent planes. Verifying that points lie on a surface is merely a matter of making substitutions.

with(Student:-VectorCalculus):
BasisFormat(false):

Define the surface in spherical coordinates as a way of avoiding the division-by-zero errors one gets in Cartesian coordinates. (An example on the help page for the TangentPlane command sparked this insight.)

S:=Vector(<2,s,t>,coords=spherical);

Convert the points to spherical from Cartesian coordinates:

A:=simplify(MapToBasis(<1,1,sqrt(2)>,spherical)):
B:=simplify(MapToBasis(<sqrt(2),sqrt(2),0>,spherical)):
C:=simplify(MapToBasis(<2,0,0>,spherical)):
A,B,C

The tangent planes are given parametrically (parametric expressions are stacked in a vector) by the following commands:

simplify(TangentPlane(S,s=Pi/4,t=Pi/4));
simplify(TangentPlane(S,s=Pi/2,t=Pi/4));
simplify(TangentPlane(S,s=Pi/2,t=0));

Write the "name" that you want to use on the left side of the assignment, select it, and convert it to an Atomic Variable.

The conversion can be done as follows:

Format Menu/Convert To/Atomic Variable (Note the keyboard shortcut keys - Control+Ahift+A)

To see that you actually have created an Atomic Variable, select "Atomic Variables" in the View Menu. Atomic Variables will then display in a magenta color. Unfortunately, each time you want to refer to this variable, you have to re-create it, or copy/paste it. 

I just experimented with the following. It might work for the OP.

alias(t'=q)

The symbol t' should be made into an atomic variable. (I select the expression and use the keyboard: Control+Shift+a. Alternatively, Select "Convert To" from the Format menu and pick the Atomic option.)

Upon input, use f(x,q), and this will echo back as f(x,t'). Of course, the alias step could be skipped, and t' could be set as an atomic variable each time it is to be entered. As this could be tedious, choose your path of least annoyance.

Another possibility is to assign the equivalent of the atomic form of t' to a name. To discover the correct atomic form of the atomic variable, execute lprint(t'), where t' is set as atomic. The echo in this case will be `#mrow(mi("t"),mo("&apos;"))`

Make an assignment such as T := `#mrow(mi("t"),mo("&apos;"))`, and the input f(x,T) will echo back as f(x,t').