[Geometry] Distinct Right Triangles
By: Tao Steven Zheng (鄭濤)
【Problem】
This is an original problem I created in 2015; it was inspired by a problem from the ''Jiuzhang Suanshu''.
How many distinct right triangles with integer sides circumscribe a circle of radius 10? List all the triangles.
Note:
by distinct I mean we do not count any permutation of . For example, the Pythagorean triple?(12, 5 ,13) is the same triangle as (5, 12
,13).
【Solution】
We use one of the formulas relating the radius of an incircle to the three sides of the right triangle , where
is the radius,
are the two legs, and
is the hypotenuse.
Since , we get
Thus,
Impose the condition , so that we only need to check the values
.
The only integer solutions for are
.
Hence, there are 6 distinct triangles:
標(biāo)簽：
