One way to find the equation of an ellipse circumscribed around a triangle. In this case, we solve a linear system of equations, which is obtained after fixing the values of two variables ( t1 and t2). These are five equations: three equations of the second-order curve at three vertices of the triangle and two equations of a linear combination of the coordinates of the gradient of the curve equation.
The solving of system takes place in the ELS procedure. When solving, hyperboles appear, so the program has a filter. The filter passes the equations of ellipses based on by checking the values of the invariants of the second-order curves.
FOR_ELL_ТR_OUT_PROCE_F.mw  ( Fixed comments in the text  01, 08, 2020)

An attempt to find the equation of an ellipse inscribed in a given triangle. 
The program works on the basis of the ELS procedure.  After the procedure works, the  solutions are filtered.
ELS procedure solves the system of equations f1, f2, f3, f4, f5 for the coefficients of the second-order curve.
The equation f1 corresponds to the condition that the side of the triangle intersects t a curve of the second order at one point.
The equation f2 corresponds to the condition that the point x1,x2  belongs to a curve of the second order.
Equation f3 corresponds to the condition that the side of the triangle is tangent to the second order curve at the point x1,x2.
The equation f4 is similar to the equation f2, and the equation f5 is similar to the equation f3.
FOR_ELL_ТR_PROCE.mw
For example

One of the forums asked a question: what is the maximum area of a triangle inscribed in a given ellipse x^2/16 + y^2/3 - 1 = 0? It turned out to be 9, but there are infinitely many such triangles. There was a desire to show them in one of the possible ways. This is a complete (as far as possible) set of such triangles.
(This is not an example of Maple programming; it is just an implementation of a Maple-based algorithm and the work of the Optimization package).
MAX_S_TRIAN_ANINATION.mw

A way of cutting holes on an implicit plot. This is from the field of numerical parameterization of surfaces. On the example of the surface  x3 = 0.01*exp (x1) / (0.01 + x1^4 + x2^4 + x3^4)  consider the approach to producing holes. The surface is locally parameterized in some suitable way and the place for the hole and its size are selected. In the first example, the parametrization is performed on the basis of the section of the initial surface by perpendicular planes. In the second example, "round"  parametrization. It is made on the basis of the cylinder and the planes passing through its axis. Holes can be of any size and any shape. In the figures, the cut out surface sections are colored green and are located above their own holes at an equidistant to the original surface.
HOLE_1.mwHOLE_2.mw

It can be considered as continuation of these topics: “Determination of the angles of the manipulator with the help of its mathematical model. Inverse problem.”  and  “The use of manipulators as multi-axis CNC machines”.
There is a simple two-link manipulator with three degrees of freedom. We change its last link with a fixed length to a variable length link. By this action we added one degree of freedom to our manipulator. Now we have a two-link manipulator with 4 degrees of freedom.
In a particular mathematical model the variable length of the link is related to the x7, and in the figure the total length of the last link is displayed as the result of subtracting the fixed part of the link (L2) from x7 (equation f2). At the same time, the value of the variable x7 depends on the inclination of the first link (equation f4).  x1, x2, x3 are the coordinates of the moving point of the first link, x4, x5, x6 are the coordinates of the operating point.  The  equation f2 defines the type of connection between the links, the equations f5 and f6  are the trajectory of the working point.

MAN_2_4_Variable_length_MP.mw