SOLUTION: A circle inscribed in 3-4-5 right triangle. How long is the line segment joining the tangency of the 3-side and the 5-side?

Question 1081655: A circle inscribed in 3-4-5 right triangle. How long is the line segment joining the tangency of the 3-side and the 5-side?
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
Let ABC be a right triangle, with AB=3, AC=4, BC=5,
Let the center of the inscribed circle be point O,
and let the points of tangency on AB and AC be M and N respectively.
We want to find the length of segment MN.
The radius of the circle is MO=NO=PO.
The area of triangle ABC is
.
Angle bisectors AO, BO, and CO split triangle ABC into triangles ABO, BCO, and ACO.
The areas of those smaller triangles are:

The areas of ABO, BCO, and ACO add up to the area of ABC, so

--->
The angles at M, N, and P are right angles,
because tangents to a circle are perpendicular to the radius at the point of tangency.
That means that we have a bunch of small right triangles (such as MOB and NOB).
It also means that AMOP is a square, with .
So, .

Triangle NOB is congruent with triangle MOB,
so , and MB=NB.
Triangles MOB and NOB together form kite MBON, with .
The area of a kite can also be calculated as half the product of the diagonals.
So, , , and
In triangle MOB, the Pythagorean theorem tells us that

Substituting into , we gwt
--> .
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