The One-Two Right Triangle
Note: we have expanded this piece with more examples, for publication in a Trader's Journal article.
One leg of this triangle is twice the length of the other leg. Therefore we refer to it as the "one-two right triangle".
This simple geometric figure can, and will, teach us many things. The first thing we'll do is put some values on its sides. Next, we'll look at the various ratio relationships between the sides and combinations of sides.
The hypotenuse is easily determined with the Pythagorean theorem, which states that the hypotenuse of a right triangle is equal to the square root of the sum of the squares of the other two sides.
Now let's look at the various ratio relationships between the triangle's perimeter, hypotenuse and sides.
Here is triangle ABC:
S&P 500 bar chart, 1987-2000
Below is a monthly bar chart of the S&P 500 covering the period from the October, 1987 "crash" low to the 2000 tops. Some of the trading day distances (number of trading days) between the 1987-1990-1994 BalancePoint lows and the January-March 2000 tops are shown on the chart. Click on it to see it full-size:
During the last three months of 1994 the DJIA and the S&P formed their last major consolidation area prior to lifting off for the meteoric rise toward historic highs in 2000. The DJIA made its intraday low for this period on November 23, the S&P on December 4. This created a CompoundPivot and BalancePoint. Daily bar charts for both indices are shown below, illustrating the derivation of the BalancePoint.
The 600-1200 Right Triangle
Fig. 1
The legs of this triangle total 1800 (600 and 1200), representing the 1800 day time distance from the 1987 low to the 1994 BalancePoint. This generates a hypotenuse of 1341.6.
The perimeter of the triangle is 3141.6. This is π scaled (multiplied by 100):
The time distance from the 10/20/87 low to the 03/24/00 S&P top is exactly 3141 days. The hypotenuse of this triangle, 1341.6, is the time distance from the 94 bottom BalancePoint to the 03/24/00 top.
Downscaling the hypotenuse value, 1341.6, by a factor of 100 gives us the ratio 1.3416. Please recall that in any one-two right triangle 1.3416 is the ratio of the sum of the legs' value divided by the hypotenuse value.
Downscaling 1800 by the same factor gives us the ratio 1.8. The square root of 1.8 is 1.3416.
Figure 1 illustrates the importance of the CompoundPivot/BalancePoint Principle. Without the BP there would not be an 1800 day move distance. Figure 1 also illustrates some interesting relationships between ratios and market moves. Most important of all, it illustrates how an important market move can be geometrically related to a future move.
More About the Importance of "1800".
Let's take another look at the One-Two Triangle with sides of 600 and 1200.
Downscaling 1,800,000 (dividing by 1000) = 1800, the sum of the sides. This "equality" between the sum of the sides and the sum of their squares is somewhat unique.
a. 
b. 
This is interesting inasmuch as it equals 3 x 1.4142, the sq. rt. of 2.
c. The inverse of 1.8 = .5555
d. The number of days between the October, 1987 low and the January, 2000 DJIA top = 3093 days.
e. 
f. August 9 and August 12 create a three day CompoundPivot at the 1982 bottom.
The number of days between 08/12/82 and 10/20/87 = 1312 (e. above)
g. The number of days between 1982 bottom BP and the 08/13/04 S&P low is 5555 (c. above).
h. The number of days between the 1987 bottom and the 08/13/04 S&P low is 4242 (b. above).
Above are just some of the ways in which "1800" is involved with major market moves; using its inverse, its square root, and also scaling 1800 and then deriving its square root. Of course, 1800 is also important in geometric relationships.
1047-2094 Right Triangle
On the monthly bar chart of the S&P, the time distance between the 1990 low and the 1994 bottom BalancePoint is shown; 1047 days. The time distance is the same for the DJIA.
Placing 1047.2 on the short leg of the One-Two Right Triangle makes the long leg 2 × 1047.2 = 2094.4. The sum of the legs is 3141.6, or π × 1000.
The hypotenuse is 2341.6. Please recall from the beginning of this article that in any One-Two Right Triangle, the perimeter over the hypotenuse=2.3416.
The perimeter of this triangle is 5483.
Therefore, the square of the hypotenuse, (2341.6 × 2341.6)&divid;1000, is the perimeter of the triangle.
The time distance from the 10/11/90 low to the DJIA 01/14/00 top is 2340 days. This is an error factor of only one day, if 1047 and 2094 are the legs used to generate the hypotenuse 2341.
If we assume a move of 2341, one more trading day than the actual 2340 day move, the time distance from the BalancePoint to this assumed day wouild be 1294 days; 2341 minus 1047. We are using this "assumption" to illustrate additional close relationships between the two 2000 tops and the 1800-day '87-'94 BalancePoint move.
The Phi Connection
The Golden Mean is almost always present in market moves. The One-Two Right Triangle is an important element in geometrically illustrating the Golden Mean. For now we'll merely take a quick look at the 2.3416 ratio.
Of course you recognize 3.236 as twice 1.618. In case 1.382 does not look familiar, remember that 0.618 squared = 0.382, and 1.382 × 1.618 squared = 3.618, and so on.
1800 and the October, 2002 Bottom
There are 1273 days between the August 12, 1982 bottom and the August 25, 1987 top.
4 × 1273 = 5092
Using both the 08/09 and 08/12 1982 bottoms there are 5091 and 5094 days between the 1982 bottoms and the October 10, 2202 bottom.
The diagonal of a square is 1.4142 (sq. rt. of 2) x one side of the square.
A square with sides of 1273, perimeter of 5092, has a diagonal of 1800.3, 1.4142 x 1273.
Figure A.
The diagonal is .35355 x the perimeter of the square, 1.4142 divided by 4.
The inverse of .3535 = 2.8284, or 2 x 1.4142.
The relationship of 1800 diagonal to the 5092 perimeter may also be graphically shown as below in Fig. B:
1800 is one side of square ABCD; 5092 is the diagonal CG of square CEFG with sides of 3600, CD, twice the size of square ABCD.
Figure B.
Now let's come full circle (not an apt phrase when working with triangles and squares) back to the one-two right triangle:
In Fig. C the diagonal of 1800.3 from Fig. A is now the short leg of one-two right triangle ABC. Triangle ADC is an isosceles right triangle (two equal legs), and the 5092 perimeter of the square, Fig. A, is the hypotenuse.
Figure C.
You may think all the above merely says the same thing in different ways. You would be correct. However, visualizing market moves in various geometric configurations, no matter how simple, is a first step toward understanding more complex market structures.
We've seen how the 1800 day move is related in simple and significant ways to many other important market moves.
What is most important to glean from this is that there would not be an 1800 day move to work with, were it not for the CompoundPivot and BalancePoint principles.