If we trisect the angles of a triangle and consider the three intersection points of corresponding trisector lines we always get an equilateral triangle (Morley's triangle)

Each point in the circle circunscribed to a triangle give us a line (Wallace-Simson line)

Interactive 'Mostration' of the Wallace-Simson line.

All the Wallace-Simson lines form a deltoid (Steiner Deltoid).

Steiner deltoid is a hypocycloid related with the nine point circle of a triangle.

Steiner Deltoid and the Morley triangle are related.

We can see Pythagoras' Theorem in a tiling. It is a graphic demonstration of Pythagoras' Theorem we can see in some floor made using squares of two different sizes.

Central angle in a circle is twice the angle inscribed in the circle.

Interactive 'Mostation' of the property of central and inscribed angles in a circle. Case I: When the arc is half a circle the inscribed angle is a right angle.

Interactive 'Mostation' of the property of central and inscribed angles in a circle. Case II: When one chord that forms the inscribed angle is a diameter.

Interactive 'Mostation' of the property of central and inscribed angles in a circle. The general case is proved.

A Dilative Rotation is a combination of a rotation an a dilatation from the same point.

He studied transformations of images, for example, faces.

In this painting we can see, among lots of interesting things, an anamorphosis of a skull. (In Spanish)

In an equiangular spiral the angle between the position vector and the tangent is constant.

Two transformations of an equiangular spiral with the same general efect.

The diagonal of a regular pentagon are in golden ratio to its sides and the point of intersection of two diagonals of a regular pentagon are said to divide each other in the golden ratio or 'in extreme and mean ratio'.

From Euclid's definition of the division of a segment into its extreme and mean ratio we introduce a property of golden rectangles and we deduce the equation and the value of the golden ratio.

A golden rectangle is made of an square and another golden rectangle.

A golden rectangle is made of an square an another golden rectangle. These rectangles are related through an dilative rotation.

Two equiangular spirals contains all vertices of golden rectangles.

The golden spiral is a good approximation of an equiangular spiral.

There is a standarization of the size of the paper that is called DIN A. Successive paper sizes in the series A1, A2, A3, A4, and so forth, are defined by halving the preceding paper size along the larger dimension.

Transforming a circle we can get an ellipse (as Archimedes did to calculate its area). From the equation of a circle we can deduce the equation of an ellipse.

Every ellipse has two foci and if we add the distance between a point on the ellipse and these two foci we get a constant.

The volume of a tetrahedron is one third of the prism that contains it.

Special sections of a tetrahedron are rectangles (and even squares)

Howard Eves, mathematician and historian of Mathematics, received the George Polya Award for the article Two Surprising Theorems on Cavalieri Congruence

We want to calculate the surface area of sections of a sphere using the Pythagorean Theorem. We also study the relation with the Geometric Mean and the Right Triangle Altitude Theorem.

We show a sphere and the Howard Eves's tetrahedron with congruent sections.

One eighth of a regular dodecahedon of edge 2 has the same volume as a dodecahedron of edge 1.

One eighth of a regular dodecahedon of edge 2 has the same volume as a dodecahedron of edge 1.

The volume of an octahedron is four times the volume of a tetrahedron. It is easy to calculate and then we can get the volume of a tetrahedron.

The twelve vertices of an icosahedron lie in three golden rectangles. Then we can calculate the volume of an icosahedron

The truncated octahedron is an Archimedean solid. It has 8 regular hexagonal faces and 6 square faces. Its volume can be calculated knowing the volume of an octahedron.

These polyhedra pack together to fill space, forming a 3 dimensional space tessellation or tilling.

We can cut in half a cube by a plane and get a section that is a regular hexagon. Using eight of this pieces we can made a truncated octahedron.

Using eight half cubes we can make a truncated octahedron. The cube tesselate the space an so do the truncated octahedron. We can calculate the volume of a truncated octahedron.

A cuboctahedron is an Archimedean solid. It can be seen as made by cutting off the corners of a cube.

A cuboctahedron is an Archimedean solid. It can be seen as made by cutting off the corners of an octahedron.

The compound polyhedron of a cube and an octahedron is an stellated cuboctahedron.It is the same to say that the cuboctahedron is the solid common to the cube and the octahedron in this polyhedron.

The stellated octahedron was drawn by Leonardo for Luca Pacioli's book 'De Divina Proportione'. A hundred years later, Kepler named it stella octangula.

We study different prisms and we can see how they develop into a plane net. Then we explain how to calculate the lateral surface area.

Plane nets of prisms with a regular base with different side number cut by an oblique plane.

We study different cylinders and we can see how they develop into a plane. Then we explain how to calculate the lateral surface area.

We study different cylinders cut by an oblique plane. The section that we get is an ellipse.

Plane net of pyramids and pyramidal frustrum. How to calculate the lateral surface area.

Plane net of pyramids cut by an oblique plane.

Plane developments of cones and conical frustum. How to calculate the lateral surface area.

Plane developments of cones cut by an oblique plane. The section is an ellipse.

Several pages about simple techniques for building polyhedra: cardboard, origami, tubes, zome, tensegrity.