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A Math quiz - How to find the area of a triangle from its incircle

– 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

Image credit Theodor Pan/Quora

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

Image credit Theodor Pan/Quora

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].