– area of a triangle A = ½ s*r where s is the sum of the 3 sides' lengths, r is the radius of its incircle
We will need to learn about incircle (the largest circle that [tangentially] touches all 3 sides of the triangle)
Note: in English ‘in circles’ (also ‘in a circle’) means ‘go around without making progress’
We can work out the area of any triangle by adding a few more lines, like this
Now, the area of the triangle should obvious as stated in the beginning.
--area of a triangle A = ½ s*r where s is the sum of the 3 sides' lengths, r is the radius of its incircle.
This example shows how ‘imagination’ is used to solved a problem - logically. We can doodle up the figures above without a ruler and a compass. We can visually see the solution. The numerical ‘answer’ can be calculated if the [data on] lengths of the triangle are available.
To draw up the incircle for a triangle exactly is another good exercise. We will need a ruler and a compass to do that. To put the question another way:
-- how to find the centre of the incircle of a triangle --by using only straight edge and compass?
[see https://www.mathopenref.com/constincircle.html on how to draw the incircle then https://en.wikipedia.org/wiki/Incircle_and_excircles_of_a_triangle for more].