# DISTANCE BETWEEN IN-CENTRE AND CIR-CUM-CENTRE.

Hello math bugs(๐) & hivers(๐)

I hope you are strong and stout and doing well.

Today the question is finding **distance between In-centre and Cir-cum-centre**.Check the following the following figure and try to find out the solution.You need to every inch and out of In-circle and cir-cum-circle and their relation to solve it.

**The distance between two above mention centres represents by :**d = โ(Rยฒ-2rR) Where **R** represents Cir-Cum-Radius and **r** represents In-Radius.

**It is time to prove.It may be quite complicated but give it a try.**

**Need to know three things:**

๐ฏWhen the line joining In-centre and the vertices is produced to Circumference of the Cir-cum-circle, what we get check below:When we try proving something in geometry, we should know related proven staff.So I am not proving it here.๐คช I'll keep it for some other day otherwise it will be very irritating to get all the things at the same time.

๐ฏ๐ฏAs **AI** in the figure is angle bisector so angle A get divided into **a** And thus angle B to **b** each.Again angle BIQ becomes **(a+b)** as external angle of a triangle is equals to sum of internal opposite angle of a โ.So we have **BQ=IQ.Check details below:**

๐ฏ๐ฏ๐ฏThe two pink and red triangles are similiar because they have two equal angles.Find it in the following figure:

**So what we get is in the following figure and also coming to the proof.Bang! Bang! here it is:****We can proof the formulla using sine rule and cosine rule also but before that I had to prove them also.If used that method it would be more complicated. So I decided use geometry.**

The In-radius is perpendicular drawn form the centre to the sides of the triangle and the Cir-Cum-Radius is distance between vertices and Cir-cum-centre. So the solution is given below.

**The construction may be in appropriate because it is done without taking measurement (what eye sight tells).**

โ Check previous related post below:

I hope you enjoyed giving a little bit trouble to your brain.Thank you so much for visiting.

Have a great day

All is well

Regards: @meta007

I definitely did not learn this in high school. This is way more advanced than I remember.

!discovery 37

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_{Please contribute to the community by upvoting this comment and posts made by @indiaunited.}Well written,although i knew it earlier. But i have one question.

How do you draw those pictures which you share in your post?

It's just my curiosity, nothing else. So giving answer is not mandatory ๐คฃ.

Hello bro

I use a math editor and text editor and both of them have option for coloring line segment and area.

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Quite descriptive ..easy to understand .โจ๐

Nop! I tried again, and fail again! ๐

No worry it a difficult one. It took me time to learn. Though proof is not important , just knowing the formulla is okay. We never ask in school why ฯ comes in area of a circle.lol

Thanks for visiting or supporting.

Well, now I feel a bit better! ๐

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