Distance of centroid to incenter

Question

Suppose there is a right triangle where all side-lengths are integers. The distance from the circumcenter to the centroid of the triangle is 6.5.

Find the distance from the centroid to the incenter of the right triangle.

Answer 1

I have found the solution to this problem.

First draw triangle ABC with AB being the hypotenuse. The circumcenter O is the midpoint of the hypotenuse. Centroid G is 2/3 of the way from right angle C to point O.

CO is now 19.5 and AB is 39. Pythagorean triples makes the legs 15 and 36 in length. With C being the origin, A becomes (0, 15), and B = (36, 0). O is now (18, 7.5) and G is (12, 5).

If we name the incenter of the triangle I and the inradius r we know that 15-r+36-r=39. Hence, r = 5 and I is (6, 6). Thus, by distance formula, GI is square root of 37.

Answer 2

Hint: Where exactly is the circumcenter of a right triangle located ?

Answer 3

HINTS:

• Find a characterization of the lengths of the three sides of the right triangle with integral sides. Search for "Pythagorean Triple." You will have expressions in three integral parameters.
• Place three convenient points on the Cartesian plane that define such a triangle. It is most convenient to place the right angle vertex at the origin and the other vertices on the positive $x$ and $y$ axes.
• For that triangle, find the coordinates of the circumcenter. As you wrote, that is the midpoint of the hypotenuse.
• For that triangle, find the coordinates of the centroid. The centroid's coordinates are the arithmetic mean of the coordinates of the vertices.
• Write an expression for the distance between the circumcenter and the centroid.
• Set that expression equal to $6.5$ and solve as far as you are able. You get an equation in the three integral parameters.
• Solve that equation in integers. There are exactly two such solutions, differing only in the order of the legs, so that is basically only one solution.
• Finalize the coordinates of the centroid.
• Find the coordinates of the incenter of that triangle. You may need to search for a formula for the incenter.
• Find the distance from the centroid to the incenter.

I leave out the details since you have shown little work of your own on the problem. If you need more details, show more of your own work and tell us where you are stuck.