Geometry Answers

N ≥ 3 points lie on a plane, but not all of them lie on a straight line. Prove that there is a circle passing through three of these points and containing none of the remaining points.

Prove that if a polygon has several axes of symmetry, they all intersect in one point.

Prove that if one convex polygon lies inside another one, then the perimeter of the inner polygon does not exceed the perimeter of the outer polygon.

Sum of four unit vectors equals zero. Prove that they can be divided into two pairs of opposite vectors.

Polygon has the center of symmetry O. Prove that the sum of the distances to its vertices is minimal for point O.

The lengths of two sides of the triangle are 3.14 and 0.67. Find the length of the third side if known that it is an integer.

A circle cuts out equal chords on all sides of the quadrilateral. Prove that it is possible to inscribe a circle in that quadrilateral.

The radius of the inscribed circle of the triangle equals 1, heights are integers. Prove that the triangle is equilateral.

M is the middle of the side AB of the triangle ABC. Prove that CM = AB/2 only if the angle ∠ACB = 90°.

The distance between the centers of circles R1 and R2 is d. Prove that these circles are orthogonal only if d² = R1² + R2².