See this picture of a quadrilateral.

Draw a quadrilateral with angles the same as those of this one and sides scaled by $1\frac{1}{2}$.

See this picture of a quadrilateral.

Draw a quadrilateral with angles different from those of this and sides scaled by $1\frac{1}{2}$.

The picture shows two circles with the same centre and two triangles formed by joining the centre to the points of intersection of the circles with two radii of the larger circle:

Prove that these triangles are similar.

The lines joining the circumcentre of a triangle to the vertices are extended to meet another circle with the same centre, and these points are joined to make another triangle.

Prove that the two triangles are similar.

The lines joining the circumcentre of a triangle to the vertices are extended to meet another circle with the same centre, and these points are joined to make another triangle.

Prove that the scale factor of the sides of the triangle is the scale factor of the radii of the circles.

A point inside a quadrilateral is joined to its vertices and the lines are extended by the same scale factor. Their ends are joined to make another quadrilateral.

Prove that the sides of the two quadrilateral are scaled by the same factor.

A point inside a quadrilateral is joined to its vertices and the lines are extended by the same scale factor. Their ends are joined to make another quadrilateral.

Prove that the angles of the two quadrilaterals are the same.