Diameter of the Largest Circle Inscribed in a Right Triangle

A short video explaining how to prove a few cool question in the Nine Chapters on the Mathematical Art. I hope that you guys all learned something new from this video :)

Timestamps:
00:00 Intro
00:28 First proof
01:39 Second proof
02:35 Summary

My homework problem (MATH446 at UBC):
The following problem is a generalization of problem 16 in chapter 9 of the Nine Chapters. Derive the expressions in the problem by applying only the geometric notions known to ancient Chinese mathematicians, which include congruence of areas of basic figures, similarity of triangles, the Pythagorean theorem, and basic algebraic identities such as the binomial rule, the conjugate rule, etc. Do not use calculus or analytic geometry (or the quadratic formula).

(i) Show that the diameter D of the largest circle that can be inscribed in a right triangle with legs a and b and hypotenuse c is given by D = 2ab/(a + b + c).
(ii) Also show that D may be expressed as D = a −(c −b).

Transcript:
This is a pretty neat question from Chapter 9 in the Nine Chapters on the Mathematical Art written by the Ancient Chinese civilization, “show that the diameter D of the largest circle that can be inscribed in a right triangle is given by D = 2ab/(a + b + c), or D = a - (c - b).” In this video, we will go over how exactly this was proven geometrically, without any of the modern algebra we use today.

First of all, let’s tackle D = 2ab/(a + b + c), where a and b are the sides, and c is the hypotenuse, with the points A, B, and C. We can start by taking the centre point of the circle, or O, and dividing the large triangle in three smaller triangles. And looking here, we see that the height of each triangle is just simply the radius of the circle, or r. From this, we can say that the area of the big triangle is equal to the area of those three smaller triangles, so we have an equation here, with the three letters depicting the triangle that we are referring to. We know the area of a triangle is ½ times the base and the height, so our equation can be expanded out like so. Conveniently, we can cancel out the ½ in front of all the terms and be left with this. Next, we can factor out the r from the right hand side, giving us this. Lastly, we isolate the r by dividing both sides by a + b + c, giving us the radius of the circle. To find the diameter, we just multiply the radius by 2, giving us 2ab/(a + b + c)!

Next, let’s tackle D = a - (c - b), with the same letter system as before. This time, we write out the radius first, and notice that the length of the triangle here must also be r. This means the length of the triangle here must be a - r, and b - r here. Next, we can draw a line from point O to point B, and we notice that those two triangles here are actually similar triangles! We can do the same from point O to point C, and we have two more similar triangles down here too. Since they are similar, then the side length here must also be a - r and b - r. This means that the length of C is just a - r + b - r, giving us this equation. We can rearrange the equation to isolate the r, and since we already have 2r, this means that this must be the diameter, giving us a - (c - b)!

And there we have it! We just proved geometrically that the diameter D of the largest circle that can be inscribed in a right triangle is given by D = 2ab/(a + b + c), or D = a - (c - b). This was a fun homework question I had and thought that it was very impressive that the Ancient Chinese mathematicians were able to solve this without the understanding of geometry and logic that we have today. I hope that y’all have learned something interesting today, thank you for watching, and good luck with everything!

#zeleonscience #maths #proof