Cercle inscrit

Dans un article précédent, nous montrions comment écrire une description textuelle du cercle circonscrit à un triangle. Cette fois-ci nous expliquons comment écrire une description textuelle du cercle inscrit à un triangle Cette description sera toujours en français, et légèrement plus complexe.

Mathematics series

My son came to me with a mathematics series, he needs to find its convergence. After the mathematics work is done, we want to confirm by calculus the found limit. Dr. Geo can be of some help here.

The series is S = Sum 1 / (k * (k + 1) * (k + 2)).
Once cut in three parts it is established it converges toward 1/4.

Then we fired up Dr. Geo to compute a few ten of thousand terms and its sum.

In a Workspace we wrote this tiny script:

| u s |
u := [ :k| 1 / (k * (k + 1) * (k + 2) ) ].
s := 0.
1 to: 100000 do: [ :k | s := s + (u value: k) ].
Transcript show: s; tab; show: s asFloat; cr

...and we got the confirmation of our finding:

(1250037500/5000150001) 0.2499999999500015

What is interesting is the exact result Pharo gave us as a fraction number. That's why we also asked for a float version of the sum to more easily realize how close we were to 1/4.

Model with perpendicular bisector

How to resolve a real world problem with perpendicular bisector?

Mathematics is the tool of the mind to build simplified model of a real world problem situation. Then within its model, the mathematics toolset helps to build solutions. We present in the following paragraphs a very simple example of this process with the use of the Dr. Geo software.