Sample two remaining vertices of isosceles triangle, given some constraints

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Trigonometry question:

I want to find a way to randomly sample the coordinates of the two remaining vertices C2 and C3 of an isosceles triangle, given one initial coordinate C1. I have the coordinate of one vertex (C1), the angle between the vertex and the two remaining vertices (let's call it theta), and the distance between C1 and the two remaining vertices C2 and C3 (let's call it R: by isosceles triangle I mean, R describes both the distance from C1 to C2 as well as from C1 to C3)

How do I randomly choose two valid points C2 and C3 that satisfy the conditions of being theta degrees apart (relative to C1) and being the same distance R from C1?

The relationships I know are relevant here are:

h = R * cos(0.5 * theta) # the length of the line between C1 and the 'base' of the triangle, i.e. the line between C2 and C3

x = R * sin(0.5 * theta) # half the length of the base, i.e. the line between C2 and C3

enter image description here

julia trigonometry coordinates matlab python
1 Answers

It's quite easy to do it by choosing a random angle between Ox line and (c1, c2) line, let's call it alpha.

Here's Julia code

c1 = [0,0]
R = 5
theta = 0.3

function rotation_matrix(theta::Real)
    @. theta * [1 -1; 1 1] |> [cos sin; sin cos]
end

alpha = 2pi * rand(Float64)

c2 = R * rotation_matrix(alpha) * [1,0] + c1
c3 = R * rotation_matrix(alpha + theta) * [1,0] + c1
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