# How to fit a triangle in a circle

If all the vertices of the triangle lie on a circle, in which case it is called a refinement and circle respectively - circumscribed around it. Construct a simple triangle at a certain circle, but how to fit a triangle in a circle, if initially there was he?

### You will need:

- compass; - Paper; - Pencil; - Line.

## Instruction how to inscribe a triangle in a circle

Step 1:

For any triangle is always possible to construct the circumscribed circle, since this curve is uniquely defined by three specified points. To detect this, it suffices to assume that the triangle defined by Cartesian coordinates of its vertices. In this case, the radius and center coordinates of a circle passing through three points must be solutions of the system of three equations of second degree, with three unknowns. This system will have a unique solution, if the given points lie on a straight line (in the latter case it is not at all solutions). But the three points lie on one line, may not be vertices of the triangle, so that this case can not even be considered. Thus, the solution is known to exist.

Step 2:

To triangle inscribed in a circle it was clearly requires that its center is located at an equal distance from all three of its vertices. The problem thus reduces to finding the center of the circumscribed circle.

Step 3:

Party inscribed triangle will be the chord of the circumscribed circle. For any such exists chord radius perpendicular to it, their intersection point divides exactly half the chord. Consequently, any middle triangle a perpendicular (ie, a straight line passing through the middle of his hand and perpendicular to it) passes through the center of the circumscribed circle. It is enough to have two of the perpendicular, and their point of intersection is the center. The radius of the circumscribed circle is uniquely determined by the distance to any of the vertices.

Step 4:

The procedure for dividing the segment in half ruler and compass is, in fact, the construction of the median perpendicular. Thus, the problem of finding the center of the circumscribed circle is reduced to a division compass and straightedge triangle two sides.

Step 5:

If a given triangle - square, the center of the circumscribed circle coincides with the middle of its hypotenuse.