Surprising Geometrical Properties -- Their Investigation, Proof and Generalization

We perform dynamic investigation of two surprising geometrical properties, each of which involves additional properties. In the first task the property belongs to two regular polygons with the same number of sides and with one common vertex. It is found that all the straight lines that connect corresponding vertices of the two polygons intersect at a single point. This point the second point of intersection of the circumcircles of the polygons. In the second task the property concerns the situation in which there are two disjoint circles which are tangent to the sides of an angle, with an isosceles triangle between them which is tangent to the two circles, so that its base lies on one angle side and its head vertex lies on the second side - the sum of the radii of the circles equals the altitude of the triangle.