Distance between intersection of circle and 2 lines

207 Views Asked by Bumbble Comm At 03 Apr 2026 - 10:57

I feel like this should be simple, but I am unable to find any solutions. I am trying to find the distance between the 2 points (red line) where a circle of radius r intersects 2 lines, given the angle between the lines.

How can I find the 2 intersecting points?

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Bumbble Comm On 01 Nov 2021 - 5:26 BEST ANSWER

It's just a mathematical problem, first you can find the slope of your lines using (respectively for degrees and radians):

$m=\tan\left(\frac{\alpha}{2\cdot180}\pi\right)$

$m=\tan\left(\frac{\alpha}{2}\right)$

Then, writing down the equations for the intersections of the lines and (half of) the circle, you obtain: $m=m^{2}r^{2}-m^{2}y^{2}=q^{2}%2By^{2}-2qy)$

Solving the equation imposing to have a unique solution to have the lines as tangents you obtain the intercept:

$q=r\sqrt{1%2Bm^{2}})$

and then:

$y_{1}=\frac{q}{1%2Bm^{2}}$

The y of the lower point is just -y1, so the overall distance is, after some simplifications:

$2r\cos\left(\frac{\alpha}{2\cdot180}\pi\right)$ (Degrees)

$2r\cos\left(\frac{\alpha}{2\cdot180}\pi\right)$ (Radians)

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Distance between intersection of circle and 2 lines

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