Ryan Morgan has gotten used to sharing trade secrets with older and wiser mathematicians. So, the Patapsco High School junior was unfazed yesterday when he explained his findings -- widely considered a geometry theorem -- during a workshop at the annual conference of the Maryland Council of Teachers of Mathematics in Ellicott City.

It was further proof that the work of a persistent student from Dundalk has captured the attention of math enthusiasts here and across the nation.

Ryan shared the stage with "my colleagues" from Towson State University's math department and his former geometry teacher. But he was the only high school student presenting information to the 2,000 teachers choosing among nearly 200 workshops.

Ryan still considers his finding a "conjecture," that is, an unproven observation, because he hasn't proven it. But others have provided proof, which is everything in geometry. Proof is what makes other mathematicians recognize a theorem as a theorem -- by showing that it can be replicated.

And others who have examined his s findings on triangles unequivocally consider Morgan's Conjecture to be Morgan's Theorem -- no small achievement for a mathematician of any age or stature.

"It's great. I love it," said Ryan, as he sat watching Towson State math Professor Robert Hanson expound on his work. "I'll be interested to see in 20 years how far it's gone."

Dr. Hanson presented not one, but four proofs -- his own; one from a mathematician turned real estate agent in Elkins Park, Pa.; a third from a professor at St. John's University in Jamaica, N.Y.; and a fourth that combines Dr. Hanson's with two other formulas.

"Without Ryan, this would not have been possible," the professor said of the work in front of him.

After Ryan's findings were published in math journals and newspapers across the nation late last year, mathematicians and engineers responded with letters, questions and proof. Letters came not only to Ryan and Dr. Hanson, but also to Ryan's former math teacher, Frank Nowosielski, who got the whole thing started two years ago when he asked his gifted-and-talented math class to prove "Marion's Theorem."

Ryan did that -- and went on to discover his own. The basics are this:

* 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 vertexes (corners or angles), there will always be a hexagon in the interior of the triangle, and

* The hexagon's area will always be a predictable fraction of the triangle's area. The fraction is determined by a complex formula that Ryan developed.

In the mathematics world, theorems are important for their own sake. They will not cure cancer or stop global warming, but they are the building blocks of other findings and a tease to some mathematicians.

"There are people who just love to do this [prove conjectures]," said Dr. Hanson. "It's a fun thing and it's certainly a challenge."

Ryan's theorem has been a challenge for him. He completed a proof last summer, but Mr. Nowosielski discovered errors in it. "I'm this close," to finding a proof, Ryan said, holding his thumb and forefinger about a half-inch apart.

This year, he's into calculus and is planning to be an engineer. "I've set my sights on MIT [Massachusetts Institute of Technology]," he said. "But I know that depends on more than mathematics."