A point at equal distances

You recreate a figure called 'House'. All the lines forming its border are 5 cm long.

• At the bottom is the floor DE, 5 cm.
• From D and E, two walls go straight up, DB and EC, each 5 cm, at right angles to the floor.
• On top is a pointed roof: two lines, AB and AC, each 5 cm, meeting at the top point A.
• Between B and C is a curve that dips down into the house. (There is also a small door, 1 cm wide and 2 cm tall.)

Step 1 is easy: draw the floor and the two walls. But how do we find A? It must be 5 cm from B AND 5 cm from C. A ruler means trial and error. Instead, remember that all the points 5 cm from B form a circle of radius 5 cm with centre B.

• Step 2: draw it. A is somewhere on this circle.
• Step 3: draw another circle of radius 5 cm with centre C.

The point where the two circles cross is 5 cm from B and 5 cm from C. That is A! You do not need whole circles: two arcs of radius 5 cm, from B and from C, crossing above the walls, are enough.

Then join A to B and A to C. For the curve between B and C, use the fact that A is 5 cm from both: Step 4, with the compass set to 5 cm and its point on A, draw the arc from B to C. The house is ready!

The book then asks: is there a four-sided figure with all sides equal that is not a square? Its hint: draw AB = 5 cm and AC = 5 cm, meeting at A at an angle that is not a right angle. The fourth corner D must be 5 cm from both B and C, so, just as with the house, draw arcs of radius 5 cm from B and from C and take the point where they cross (on the far side from A). The figure ABDC has all four sides 5 cm, but its angles are not 90°, so it is not a square.