The Art and Craft of Problem Solving

270 CHAPTER 8 GEOMETRY FOR AMERICANS

I. Start with a long strip of paper, and visualize

a crease for folding (thin line). The crease can

have any angle.

2. Hold the top left comer and fold it down on this
   crease so that the comer is now below the bot­
   tom of the strip.


3. Unfold. You now actually have a crease (shown
   by thin line).


4. Now grab the right end of the strip and fold
   DOWN so that the top side of the strip is along
   this crease.


5. Unfold. You now have two creases.


6. Now grasp the right end and fold UP so that the
   bottom side of the strip is along the most re­
   cently created crease.


7. Unfold. You now have three creases.

(^8). Repeat steps 4--7. Your creases will now be
equilateral triangles!
Comment on this procedure. Does it work? Does it
almost work? Explain!
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8.2.34 (Bay Area Mathematical Olympiad 1999) Let

C be a circle in the xy-plane with center on the y-axis

and passing through A = (O,a) and B = (O,b) with

° < a < b. Let P be any other point on the circle, let Q

8.3 Survival Geometry II

Area

be the intersection of the line through P and A with the

x-axis, and let 0 = (0,0). Prove that LBQP = LBOP.

8.2.35 (Canada 1991) Let C be a circle and P a given

point in the plane. Each line through P that intersects

C determines a chord of C. Show that the midpoints of

these chords lie on a circle.

8.2.36 (Bay Area Mathematical Olympiad 2000) Let

ABC be a triangle with D the midpoint of side AB, E

the midpoint of side BC, and F the midpoint of side

AC. Let kl be the circle passing through points A, D,

and F; let k2 be the circle passing through points B, E,

and D; and let k3 be the circle passing through C, F,

and E. Prove that circles k I , k2' and k 3 intersect in a

point.

8.2.37 Let ABC be a triangle with orthocenter H (re­

call the definition of orthocenter in Example 8.2.23).

Consider the reflection of H with respect to each side

of the triangle (labeled HI ,H2,H 3 below).

• HI

B

Show that these three reflected points all lie on the cir­

cumscribed circle of ABC.

We shall treat the notion of area intuitively, considering it to be "undefined," like points

and lines. But area is not an "object," it is a/unction: a way to assign a non-negative

number to each geometric object. Everyone who has ever cut construction paper has

internalized the following axioms.

• Congruent figures have equal areas.


• If a figure is a union of non-overlapping parts, its area is the sum of the areas of
  its component parts.


• If a figure is a union of two overlapping parts, its area is the sum of the areas of

the two parts, minus the area of the overlapping region. This is the geometric