Explorations in Geometry

with the

Exploration 21.

In this exploration you will examine a possibly-new theorem of triangles discovered by Harold Stengel, and proven by him and a colleague. Along the way we will try to develop some intuition about triangles and their cevians in general. To assist you, I have led you to a slightly different version of the theorem, which I hope is a little easier to discover. Mr. Stengel to sent me a paper about his discovery. Here is the problem which led him to his new theorem.

1. On your GEOMETER'S SKETCHPAD create a horizontal line, and then select two points on the line to serve as the base of a triangle (B and C above) (Make sure to do this step, it will make some of the later steps easier to do--- don't just create a line segment for the base)

2) Select another point OFF the line to serve as the third vertex of the triangle. Then connect them to form a triangle. (You may want to create all this and then rename the vertices using the text pointer).

3) Now select a point on sides AC and AB (F and G above) and connect them to the opposite vertex to form a cevian. Name another point at their intersection (H above)

4) Now measure the perpendicular distance of each of these three points (F, G, and H) from the base of the triangle. (BC). I will refer to these lengths with the lower case letters f, g, and h.

5) Move points B and C back and forth along the original line and observe the impact on the lengths of the three measurements. What do you observe?

6) Now select the other vertex of the triangle (A) and move it closer and farther from the line while observing the three lengths.

7) Now one at a time, move the the points F and G along the sides of the triangle and observe the lengths f, g, and h.

8) The discovery by Mr. Stengel involved a relationship between these three lengths, can you find it. Be warned, it is not easy to see. If you are really stuck, scroll way down to the bottom of the page and read Hint number 1.

9) One of the relationships I hope you found was that the three lengths could be written so that they formed a constant for a given triangle. This constant would not change if you moved points F and G along the sides. (If you are still stuck, look at Hint number 2, which is really an answer)

10) There was another length involved in Mr. Stengel's discovery, a length that is related to the constant you found in part 9). Can you find the length? Hint number three has the solution.

11) If you found all of these, you can use the four lengths into a single equation, and it was in such an expression that Mr. Stengel stated his discovery. Can you write it? Hint number three has his expression. Test the theorem on the sketchpad by calculating the values of each expression. Go back and move lots of points around and see what happens. What if you move Point A?

|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|

HINT NUMBER 1……..try looking at the reciprocals of the lengths. Examine various sums and differences to see if any of them look interesting when you move stuff around.

|
|
|
|
|
|
|
|
|
|
|
|
|
|
HINT NUMBER TWO…..1/f+1/g-1/h is a constant value if you move B,C, F or G, even though all the values may change.

|
|
|
|
|
|
|
|
|
|
HINT NUMBER THREE---the length is the height. The constant in the previous parts was the reciprocal of the height. So one way to write an equation would be:

1/f+1/g-1/h = 1/a where a is the height from A to the base, the height of the triangle. Mr. Stengel posted his theorem in the following form:

1/f + 1/g = 1/a+1/h