I’ve been playing Euclidia and I love it so far. Here’s some of the interesting challenges that I was stuck at for a while. The stage which the puzzle happens is included in the brackets.

The following challenges must be solved using only elementary Euclidean operations, which is defined as a construction that can be made with a real compass and straight edge. Marking intersections are not counted as an operation. This rules out the use of protractors to measure angles.

## Constructing tangent to circle at a point (2.8)

Given a circle and a point, construct the tangent to the circle at that point with 3 elementary operations.

## Erecting a perpendicular on a line (2.7)

Given a line and a point on the line, construct a perpendicular at that point with 3 elementary operations.

## Inscribing a square (1.7)

Given a circle and a point on the circle, construct the inscribed circle with 7 elementary operations. The 4 lines are part of the operations!

# Solution

## Constructing a tangent

This works by proving that TAO1 is the same angle as ABO1. By draing a circle centered at O, we reflected B against line AO1 to T.

## Erecting a perpendicular

This uses the fact that BC is the diameter so A is right angle.

## Inscribed circle

This feels almost like magic to me.

We first draw two auxillary circles, label the intersections and construct R as follows: