Note: since I wrote this post, level 20 has moved to level 23. It may move again in the future, I suppose. Rather than keep updating, it’s the one with the tangent to two circles.

I LOVE Euclid, the game - it’s a brilliant, interactive way to get students (and their teachers) thinking geometrically - using Geogebra to check constructions without ever using a hazardous, pointy compass.

Now, I rattled through the first 19 of the 20 levels - mainly standard geometry puzzles I’ve seen often enough before. But level 20 got me thinking: how do you construct a tangent to two circles?

### My thinking

If I had a tangent to two circles, I could create two similar, right-angled triangles using the tangent, the radius of each circle from the tangent point, and the centre of each circle. In the applet below, those are HBT and HAU. So now I have two problems:

• How do I find H? and
• How do I find a point T that makes HBT a right angle?

### Finding H

If HBT and HAU are similar, the ratio of HB to HA must be the same as the ratio of the radii ((say this several times quickly)) - which means, if I can somehow construct two similar triangles of the appropriate dimensions, I’ll find H.

To do that, I can construct two parallel radii (AG and BF) and draw a line through the points on the circumference (G and F). That line cross the line AB at H.

### Making HBT a right angle

I’ve got two points (H and B) and I need to find a third point, T, on circle b, such that HBT is a right angle. How can I do that? Easy! If I construct a circle (or a semicircle) with HB as a diameter, I know the angle HBT has to be a right angle - so HT is tangent to circle b, as well as to circle a. That’s clearer from the diagram below:

[iframe http://market.flyingcoloursmaths.co.uk/geogebra/sineruleproof/m128378-Level-20.html 800 800]