How to fit a circle in a convex quadrilateral

You will need
  • Paper, pencil, ruler, compass, protractor, set square.
For the most difficult case - the construction of a circle inscribed in a quadrangle of irregular shape - it is necessary to construct the bisectors of the angles lying at the vertices of the figure. Start from any vertex - attach a protractor, measure the angle, divide the result in half and put an auxiliary point. Draw an auxiliary segment lying on the bisector of the angle of this vertex — it must start at the vertex, pass through the auxiliary point and end on the opposite side of the figure.
Repeat the previous step for the second vertexquadrangle, and at the intersection of two auxiliary segments put a dot. Mark it, for example, with the letter O - it is the center of the inscribed circle. If from the first step or from the conditions of the problem it follows unambiguously that it is possible to enter a circle in this quadrilateral, there is no need to build the bisectors of the angles in the two remaining vertices.And if the check from the first step is impossible for some reason, you should make sure that all four bisectors intersect at the same point. If this condition is not fulfilled after repeating the first step for the remaining vertices, then entercirclein such a quad is impossible.
Determine the radius of the inscribed circle. To do this, use a square or protractor to build a perpendicular dropped from the center of the circle — the point O — to either side. Put the length of the resulting segment on the compass.
Drawcirclewith the radius deposited on the compass, and the center at the point O. At this point, the construction will be completed.

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