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# SOMETHING NEW IN A TRIANGLE

Ryan Morgan would have gotten an "A" in geometry even if he hadn't unearthed a mathematical treasure.

But the persistent Patapsco 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 proved 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.

"Ryan's really done something pretty fantastic," said Mr. Nowosielski, who taught Ryan's ninth-grade geometry class for gifted and talented students last year and now teaches at the Carver Center for Arts and Technology in Towson.

"How many kids in the world have done this? He saw something and he didn't quit. He's a special kid," Mr. Nowosielski said.

What did Ryan see?

Initially, he saw a triangle, each side divided into thirds. Lines drawn from those segments to the vertices (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 Mr. 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, and

* When lines are drawn from the division points on the sides of the triangle to the vertices, 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.

Numerically, it works this way: when the sides are divided by 3, as in Marion's Theorem, the area of the hexagon will be 1/10 as large as the area of the triangle. When the sides are divided by 5, the area of the hexagon will be exactly 1/27th of the area of the triangle, according to Ryan's conjecture.

How did Ryan come to this conjecture?

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

Some would not call Ryan's pursuit a matter of "play."

After working with Marion's Theorem, "Ryan sort of said, 'That was pretty interesting,' and kept playing on the computer to see if the same concept would apply to a square and then to a five-sided figure and then a six-sided figure," Mr. Nowosielski recalled.

It didn't, so Ryan went back to the triangle.

Mr. 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.

Ryan said his parents, Jim and Hilda Morgan of Plainfield Road in Dundalk, are "proud, I guess." His classmates, too, are happy about his accomplishment.

"I don't like to brag about it because I feel stuck up," said Ryan, who plays drums in the school band, rides his bicycle to school and enjoys a normal high school life. "I feel kind of weird about it."

Ryan is also realistic about his discovery. "The only place it can be used is in math," he said. But, still, "It is really important to get it out. Maybe someone will build upon it," he said.

"To portions of the math community, it's a big deal," said Towson State's Dr. Hanson. "Considering Ryan's age and his background, it's certainly a significant contribution. There's a good deal of reasoning and time [in it]. I'm very, very pleased that a ninth-grader is willing to deal in generalities as he did -- to persevere and not to give in."

Proof is everything

Several of the Towson professors have proved the conjecture, but Ryan wants to prove it himself before looking at their versions. "It's probably a theorem now, but I still like to refer to it as Morgan's Conjecture until I personally prove it," he said.

The proof is everything in geometry -- that's what makes other mathematicians recognize a theorem as a theorem. "There's no one who stamps it and says, 'It's a theorem now,' " Ryan said.

Dr. Hanson has inquired among mathematicians in this country and in Canada to see if anyone knew about Morgan's Conjecture before Ryan Morgan. So far, he's found no one.

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

"This particular audience is really quite critical," said Joan Armistead, editorial coordinator for the journal, which is published by the National Council of Teachers of Mathematics in Reston, Va.

She said that Ryan's letter was reviewed by an editorial consultant who is a practicing mathematician before it was published and that publishing a student's letter is highly unusual.

Fame in a theorem?

"It's definitely not commonly known," Ms. Armistead said of Morgan's Conjecture. "There are probably many, many discoveries like that that can be made . . . , but usually they are made by professional mathematicians."

She added that new computer software, which draws and computes, enables students such as Ryan to do "original math."

Is there fame and fortune for someone who discovers a theorem?

"That's what my dad said, 'Where's the money in this?' " joked Ryan.