# Distance to circle inside triangle

I apologise for the lack of a precise term or title, math isn't my strong suit.

I'm trying to calculate the length of L, given angle A and radius of circle D, so that lines b and c tangents with circle D.

Can anyone help me?

Hint: Using the definition of sine and the orthogonality of the tangent one has $$\sin\left(\alpha/2\right)=\frac{\text{radius}}{\text{distance from center}}$$

• It's actually $3/\sin(\dots)$. Moreover, fortunately, since $$\sin(45/2)=\frac{\sqrt{2-\sqrt{2}}}{2},$$ you even have a closed form for the distance – b00n heT Jul 31 '19 at 13:18
• Sorry I messed up my previous comment. To repeat: So given radius 3 and angle 45 that would be $$\sin\left(45/2\right)=\frac{3}{\text{distance from center}}$$ And that isolated would be $$\text{distance from center}=\frac{3}{\sin\left(45/2\right)}$$ – Martin Jul 31 '19 at 13:22
• That's correct. – b00n heT Jul 31 '19 at 13:24
• Thank you very much. Sorry, first time formatting equations, so that took at bit. No previews for comments. :) – Martin Jul 31 '19 at 13:24
• No worries! glad to be of help – b00n heT Jul 31 '19 at 13:24

Let's call the intersection of $$b$$ with the circunference $$R$$ and $$r=d(R,D)$$, the radius of the circumference.

Considering the right-angle triangle [ARD], we have that $$\sin(RÂD)=\frac{r}{L}$$, from the definition of the $$\sin$$ of an angle.

Then, if $$A=2RÂD$$, then: $$\sin\left(\frac{A}{2}\right)=\frac{r}{L} \Leftrightarrow L=\frac{r}{\sin\left(\frac{A}{2}\right)}$$

• But in my example [ARD] isn't a right angled triangle, right? – Martin Aug 1 '19 at 11:48
• $b=AR$, $b$ is tangent to the circumference in the point $R$, therefore, $RD\perp b$, so, $\sphericalangle ARD$ is a right angle, its amplitude is 90 degrees. Note that the point $R$ is not represented in your picture, I "created" that point. So, $[ARD]$ is a right angled triangle. See tangent line in Wikipedia for more information. – t3m2 Aug 1 '19 at 12:07
• Ah yeah, makes sense. I don't know what I was imagining. It's obviously right angled. :D – Martin Aug 1 '19 at 12:11

L = R / Sin(A/2)

where R is the radius of the circle and A is the angle