Constructing with ruler and compass

Keep your geometry box ready: this chapter needs a ruler and a compass (परकार). Given a segment XY, how do we draw its perpendicular bisector using only an UNMARKED ruler and a compass? We have the idea already: join any two points, one above and one below, that are equidistant from X and Y.

Step 1: choosing some fixed radius, draw two sufficiently long arcs above XY, one centred at X and one at Y. Call their meeting point A. Step 2: with the SAME radius, draw two arcs below XY and call their meeting point B. Step 3: AB is the required perpendicular bisector.

That is all — two of the simplest tools there are, and the result is geometrically exact. It is also more accurate in practice than finding the midpoint by measuring with a marked scale, where the eye must judge a mark.

Here is a useful fact that falls out of the proof: ANY point equidistant from X and Y lies on the perpendicular bisector. So if you pick a different pair of points C and D with CX = CY = DX = DY, they must both lie on AB — because XY has only ONE perpendicular bisector. That is how you draw eyes of different shapes: choose different pairs of centres along the same line.

Now extend the method. Draw a line, mark a point O on it, and construct a 90° angle at O. The trick is to find a SEGMENT of the line for which O is the midpoint.

Using the compass, mark points X and Y on the line at equal distances from O, one on each side. Now O is the midpoint of XY, so the perpendicular bisector of XY passes through O and is perpendicular to the line.

Do we need TWO pairs of intersecting arcs this time? No — we already have one point on the perpendicular bisector, namely O. One pair of arcs gives a second point A, and joining OA gives the 90° angle. Worth exploring by construction: must the arcs above and below have the same radius? Must the two pairs be on opposite sides? Try it and justify what you find.