Can you work out how to draw a perfect right angled triangle with two equal sides, using only a straight edge and compass. That is the sort of challenge the Ancient Greek Mathematician Euclid set himself. You need to devise an algorithm to do it – a sequence of steps that guarantee you end up with a right-angled triangle.
Try it before you read on. It is a fairly simple algorithm. You need to look for a way to draw circles with points that give you a right angle.
Here is my full algorithm for drawing a square. It does give you a square, but it is hard to follow and it was just as hard to work out. I kept getting confused with which circle was which, and being sure what we did really was what I intended…what a nightmare.
- Draw a circle with radius AB centred on A, a point on a line to give a point B where line and circle cross.
- Draw a circle with radius AB centred on point B. This gives two points C and D where the circles cross.
- Draw a line through C and D. This is now at right angles to the original line. This gives a new point E where the two lines cross.
- Draw a circle with radius EC, centred on E. This gives a new point F where the circle crosses the original line in the circle.
- Draw a line CF.
- You have a perfect right angled triangle with two equal sides: CEF.
Given a right-angled triangle like this can you now see how to draw a square, starting with it (again only using a compass and straight edge).
Can you see how we could have decomposed this into two steps? (1. draw a line, perpendicular to another line 2. turn it into a triangle with 2 equal sides)
Can you work out how to construct a right-angled triangle but where its right-angle corner must be at a given point that you start with?
Can you prove (i.e. provide a totally convincing logical argument) that the algorithm works? i.e. that it is a right-angled triangle with two equal sides?