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Geometry Theory Adds Up for Baltimore Youth, 15

THE BALTIMORE SUN

Ryan Morgan would have gotten an “A” in geometry even if he hadn’t unearthed a mathematical treasure. But the persistent Baltimore high school sophomore pushed a hunch into a theory. He calls it Morgan’s Conjecture, and is hoping it will soon be Morgan’s Theorem.

In geometric circles, developing a theorem is a big deal--especially if you’re only 15.

Ryan’s teacher at Patapsco High, Frank Nowosielski, has been teaching 20 years and has never had a student discover a theorem--a mathematical statement that can be proven universally true.

Towson State University math professor Robert B. Hanson never had a high school student present a possible theorem to his faculty seminar--until Ryan did it last spring.

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“Ryan’s really done something pretty fantastic,” said Nowosielski, who taught Ryan’s 9th-grade geometry class for gifted and talented students last year.

Initially, he saw a triangle, each side divided into thirds. Lines drawn from those segments to the vertexes (the corners) formed a hexagon inside the triangle. The area of the hexagon is one-tenth the area of the triangle. This is known as Marion’s Theorem, which Nowosielski called on his class to prove as a project.

When others in the class had moved on, Ryan said, he continued to look for “something neat” in the triangle. And after lots of looking, he found that Marion’s Theorem could be extended this way:

* When the sides of a triangle are divided by an odd number larger than 1;

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* And when lines are drawn from the division points on the sides of the triangle to the vertexes, there will always be a hexagon in the interior of the triangle;

* And the area of that hexagon will always be a predictable fraction of the area of the larger triangle. The fraction is determined by a complex formula that Ryan worked out.

How did Ryan come to this conclusion?

“It’s mostly just luck in playing around with it,” he said. “I just wanted to find out something neat.”

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Nowosielski said, “It took (Ryan) a long time . . . many days after school in the computer lab,” before he started to try dividing the sides of the triangle by numbers other than three and the relationship began to emerge.

Hanson has inquired among mathematicians in the United States and Canada to see if anyone knew about Morgan’s Conjecture before Ryan Morgan. So far, he’s found no one.

In December, Ryan’s letter explaining his conjecture was published in the journal Mathematics Teacher, which goes to 60,000 high school and college math instructors.


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