18.2 Analytical Geometry
The following exercise is cited from a German textbook.
The plane E and the set of planes g_{p} mit p are given.
 Find an equation of the sphere with midpoint P(212) that has E as tangentplane.
Compute the point where E touches the shpere.
 Show that g_{p} lies in E for all values of p.
 Which lines are tangents of the sphere from part a)?
18.2.1 Solution with Archimedes

 Enter the plane via ’Typical tasks  Plane  General form’. One has to know that the
given normal form is the same as x + 2y + 2z = 18.
 Enter the point by doublerightclicking into the scene an setting the coordinates
afterwards.
 Drop the perpendicular from P to E, name the perpendicular L.
 Measure the distance from P to L. You can draw a vector or use the Macro ’Distance
 PointPoint’.
 Draw the sphere around P through L.
The sphere has the equation ^{2} = ^{2}.
 The claim can, of course, only be verified, not proven with Archimedes.
 Generate a slider via ’Extras  Slider’, name it ’pp’ (as the name P is already used,
there mustn’t be a second object of that name, even if you use the lowercase letter).
 Enter the equation of the line at ’Typical tasks  Line in parametric form’ with
the parameter pp like it is written in the exercise. Replace ’p’ with ’pp’. Select the
checkbox ’Expression contains terms’.
 If you change the value of pp with the slider you will see that the line moves within
E.
 We can find out by trying that the solution for the parameter p is 2, or we could construct the
line (as the parameter is not asked for) and find out the equation by right click 
description.