How to calculate coordinates of 6 point of intersection between a circle and a equilateral triangle? - geometry

Question

I know of an equilateral triangle the center (cx,cy) and the radius (r) of a blue circle which circumscribed it.
If I draw a green circle of any radius (radius), assuming the circle is large enough to have this intersection, can I get the coordinates of the 6 intersection points (P1, P2, P3...)?
I'm looking for P5JS/processing but any other clue can help me...
Thank you in advance

Answer 1

Distance from the center to top point is r.
Distance from the center to the lowest triangle side is r/2 (median intersection point is center, they are divided in 1:2 ratio).
Horizontal distance from cx to p4 (and p5) is (Pythagoras' theorem)
dx = sqrt(radius^2 - r^2/4)
So coordinates of p4 and p5 are (relative to center)
p4x = dx
p4y = r/2
p5x = -dx
p5y = r/2
Other points might be calculated using rotation by 120 degrees
p2x = p4x*(-1/2) - p4y*(sqrt(3)/2)
p2y = p4x*(sqrt(3)/2) + p4y*(-1/2)
and so on.
And finally add cx,cy to get absolute coordinates

Answer 2

If you want to test... ;-)
function setup() {
createCanvas(500, 500);
const cx = width / 2;
const cy = height / 2;
const r = 250; // taille du triangle
const radius = 180; // externe
noFill();
strokeWeight(3);
stroke(0, 0, 0);
drawTriangle(cx, cy, r);
strokeWeight(1);
stroke(0, 0, 255);
circle(cx, cy, r * 2);
strokeWeight(2);
stroke(8, 115, 0);
circle(cx, cy, radius * 2);
noStroke();
fill(215, 0, 0);
// dx = sqrt(Math.pow(r / 2, 2) - Math.pow(r / 2, 2 / 4));
dx = sqrt(radius * radius - (r * r) / 4);
p4x = dx;
p4y = r / 2;
circle(cx + p4x, cy + p4y, 20);
text("p4", cx + p4x, cy + p4y + 30);
p5x = -dx;
p5y = r / 2;
circle(cx + p5x, cy + p5y, 20);
text("p5", cx + p5x - 10, cy + p5y + 30);
p6x = p4x * (-1 / 2) - p4y * (sqrt(3) / 2);
p6y = p4x * (sqrt(3) / 2) + p4y * (-1 / 2);
circle(cx + p6x, cy + p6y, 20);
text("p6", cx + p6x - 30, cy + p6y);
p2x = p6x * (-1 / 2) - p6y * (sqrt(3) / 2);
p2y = p6x * (sqrt(3) / 2) + p6y * (-1 / 2);
circle(cx + p2x, cy + p2y, 20);
text("p2", cx + p2x + 10, cy + p2y - 10);
p1x = p5x * (-1 / 2) - p5y * (sqrt(3) / 2);
p1y = p5x * (sqrt(3) / 2) + p5y * (-1 / 2);
circle(cx + p1x, cy + p1y, 20);
text("p1", cx + p1x - 20, cy + p1y - 10);
p3x = p1x * (-1 / 2) - p1y * (sqrt(3) / 2);
p3y = p1x * (sqrt(3) / 2) + p1y * (-1 / 2);
circle(cx + p3x, cy + p3y, 20);
text("p3", cx + p3x + 20, cy + p3y - 10);
noFill();
stroke(0, 255, 255);
triangle(cx + p2x, cx + p2y, cx + p4x, cx + p4y, cx + p6x, cx + p6y);
stroke(255, 0, 255);
// prettier-ignore
triangle(
cx + p1x, cx + p1y,
cx + p3x, cx + p3y,
cx + p5x, cx + p5y,
)
}
function drawTriangle(cx, cy, radius) {
noFill();
trianglePoints = [];
for (var i = 0; i < 3; i++) {
var x = cx + radius * cos((i * TWO_PI) / 3.0 - HALF_PI);
var y = cy + radius * sin((i * TWO_PI) / 3.0 - HALF_PI);
trianglePoints[i] = {
x,
y,
};
}
triangle(
trianglePoints[0].x,
trianglePoints[0].y,
trianglePoints[1].x,
trianglePoints[1].y,
trianglePoints[2].x,
trianglePoints[2].y
);
}
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