The vertices of the triangle abc divide the circle circumscribed about the triangle in the ratio 2: 3: 4.

The vertices of the triangle abc divide the circle circumscribed about the triangle in the ratio 2: 3: 4. Find the corners of this triangle.

If a triangle is inscribed in a circle, then all the corners of the triangle are inscribed in a circle.
The angle inscribed in the circle is equal to half of the arc on which it rests.
Let us find the degree measures of the arcs into which the circle is divided by the vertices of the triangle.
The sum of the degree measures of all arcs is 360 °.
Let 1 part be equal to x °. Then the degree measures of arcs are 2x °, 3x °, 4x °, respectively.
Let’s compose and solve the equation:
2x + 3x + 4x = 360
9x = 360
x = 360/9
x = 40
The arcs are 2 * 40 = 80 °, 3 * 40 = 120 °, 4 * 40 = 160 °.
The angles of the triangle are 80/2 = 40 °, 120/2 = 60 °, 160/2 = 80 °.