*The greatest value of a picture is that it forces us to notice what we never expected to see. - John Tukey*

Another excellent resource is David Joyce's interactive version of Euclid's Elements.

The links below will take you to the theorems. I will (slowly!) update this page as I code more theorems. Suggestions are welcome!

- The Euler Line of a triangle
- The Nine Point Circle of a triangle
- The Simson line of a triangle
- Pappus' theorem (see also Pappus' theorem for parallel lines)
- Pascal's theorem on the circle
- Desargues' theorem (see also Desargues' theorem for parallel lines and a 3D picture of Desargues' theorem)
- Monge's theorem
- Poncelet's theorem on the circle (check out the animation of a Poncelet quadrilateral)
- The angle bisectors of a quadrilateral form a cyclic quadrilateral

- The "proof" that every triangle is isosceles
- Pythagoras' theorem from Question 6 on Tutorial 4
- Picture for Question 4 on Homework 1
- A hint for Question 2 on Homework 2

If you project a circle from one plane onto another, then the resulting figure will be a conic section. It is natural to ask how to precisely construct this conic (i.e. how to construct the focal points). The page How to draw a circle in perspective explains how to do this for the case where the conic section is an ellipse.

If you are interested in perspective drawing, then you may be interested in watching a talk about the Mathematics of Sidewalk Illusions.

- The Concentric Circle construction of an ellipse
- The Concentric Circle construction of a hyperbola
- The Trammel of Archimedes construction of an ellipse
- An explanation of how to construct an ellipse inscribed in a Symmetric Trapezoid
- An explanation of how to construct an ellipse inscribed in a kite
- Constructing a tangent to an ellipse from a point outside the ellipse.

Click here to see an illustration of the theorem that inversion preserves orthogonal circles.

The following link will take you to an explanation of the three different constructions of the inverse that we studied in class.

Here is an example of Steiner's Porism.

Click here for an interactive picture of the inverse of a pencil of parallel lines.

Click here for an interactive picture explaining Question 6 on Tutorial 9 (constructing Poncelet quadrilaterals).

In Tutorial 11 you have to do some ruler and compass constructions in the Poincare disk model of hyperbolic geometry. Below are links to interactive pictures of these constructions.

- Constructing a hyperbolic line
- Constructing a hyperbolic circle
- Constructing a hyperbolic equilateral triangle

- Constructing the circumcircle of a hyperbolic triangle
- Constructing a hyperbolic angle bisector
- Constructing the incircle of a hyperbolic triangle

The following links contain interactive pictures explaining

- The Cayley-Bacharach theorem,
- the coresidual of four points on a cubic, and
- the coresidual of seven points on a cubic.

The Poncelet-Steiner theorem says that if you are given a single circle and its centre, then any points constructible with a ruler and compass are also constructible with a ruler alone. The Poncelet-Steiner workbook explains how this works with the aid of Geogebra illustrations.