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Lazy Geometry

In high school we learn to draw angles and shapes using a straight edge and a compass. We connect points with the straight edge and marked off equal segments using the compass. Many seemingly trivial constructions require several strokes of the pencil and swings of the compass. Grading and verifying these constructions can be most time consuming. That is why high school students are never asked to model the space shuttle using only the straight edge and the compass.

The compass is not a comfortable device to maneuvre. It is inaccurate for making arcs of very small radii. On the other hand, a non-trivial amount of torque is required to make arcs of larger radii. In the true spirit of laziness, if we minimize the use of the compass, we make cleaner and more accurate drawings while reducing energy expenditure. So let us pose the following problem:

Suppose that a swing of the compass requires 5 Calories of energy, and a stroke of the pencil along a straight edge requires 1 Calorie of energy. Given a line L and a point A on L, you are asked to construct a point T such that AT and L form a 45-degree angle. How many Calories are required for this construction?


Typically, the first step is to construct a right angle. Then in the second step, the right angle is bisected, thus forming the desired 45-degree angle. The picture on the left shows this construction with minimal optimization. The steps are:
  1. Using the compass, construct a circle centered at A, intersecting line L at P and Q (5 Calories).
  2. Increase the radius of the compass. Construct two circles, one centered at P and one centered at Q, both of the same larger radius (10 Calories). These two circles intersect at R.
  3. Using the straight edge, construct line RA (1 Calorie). Line RA is perpendicular to line L, and intersects the first circle at S.
  4. Using the compass, construct a circle centered at S of the same radius as the two circles in step 2 (5 Calories). (This radius is equal to the length of RQ). This fourth circle and the circle centered at Q intersect at T. Lines AT and L form a 45-degree angle.
Total: 4 circles and 1 line requiring 21 Calories.


We can optimize this construction a little bit. After the construction of S in step 3, we already have the 45-degree angle formed by lines PS and L. If we shift our construction, we save 5 Calories. The steps are:
  1. Choose an arbitray point P on line L. Using the compass, construct a circle centered at P of radius PA, intersecting line L at A and B (5 Calories).
  2. Increase the radius of the compass. Construct two circles, one centered at A and one centered at B, both of the same larger radius (10 Calories). These two circles intersect at R.
  3. Using the straight edge, connect R and A (1 Calorie). Line RA intersects the first circle at T. Lines AT and L form a 45-degree angle.
Total: 3 circles and 1 line requiring 16 Calories.


The first method involves the construction of the perpedicular bisector followed by that of an angle bisector. The second method circumvents the construction of the angle bisector by shifting the right angle. However, there is a lazier way to construct a right angle. The steps are:
  1. Choose an arbitrary point P not on line L. Using the straight edge, construct line PA (1 Calorie).
  2. Using the compass, construct a circle centered at P of radius PA (5 Calories). This circle intersects line L at B and line PA at C.
  3. Using the straight edge, construct line BC (1 Calorie).
  4. Using the compass, construct a circle centered at B of radius BA (5 Calories). This circle intersects line BC at T. Lines AT and L form a 45-degree angle.
Total: 2 circles and 2 lines requiring 12 Calories.

By applying the fact that the hypotenuse of a right triangle equals the diameter of its circumscribing circle, we save four Calories.


Homework: What is the laziest way to construct a regular pentagon using a straight edge, a compass and a piece of paper?

Answer



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