What is the radius of the largest circle inscribed in a right angle triangle?

Answer 1

FormulaGiven a right triangle ABC, where AB is the hypotenuse, the formula for the radius of largest circle that can be inscribed is:

r = ( BC2 - BC2/AB )½ / (1+√2)

Proof">Proof">ProofIt is easy to see that the radius of the inscribed circle must touch the edges of the triangle at the hypotenuse, height, and base. Furthermore, radius lines that extend to the base (AC) and to the height (BC) of the triangle must be at right angles to one another. Finally, the radius that extends to the hypotenuse (AC) if continued outward will reach point C of the triangle, dividing the angle in half.

Let the point at which the radius of the inscribed circle touches the hypotenuse (AC) of the triangle be called Z.

This divides the original triangle into two similar triangles, CBZ and ACZ.

Applying the principle of similar triangles, we know that:

(1) BC / AB = ZB / BC solving for ZB,

(2) ZB = BC2 / AB

The length of the line ZC can be found using the Pythagorean theorem:

(3) ZC = √( BC2 - ZB2 )

However, since the radius of the inscribed circle passes through ZC we can write ZC in terms of that radius. The formula below becomes more evident when you draw a picture to illustrate the problem.

(4) ZC = r ( 1 + √2 )

We can now combine (4) and (3) to solve for r.

(5) √( BC2 - ZB2 ) = r ( 1 + √2 ) solving for r yields:

(6) r = √( BC2 - ZB2 ) / (1+√2) substituting (2) for ZB yields the final formula

(7) r = ( BC2 - BC2/AB )½ / (1+√2)

The answer makes the following statement:

Finally, the radius that extends to the hypotenuse (AC) if continued outward will reach point C of the triangle, dividing the angle in half.

This is not necessarily a true statement. It is only true for a 45 degree, 45 degree, 90 degree triangle. The formula given does not give the correct answer of 2 for a 6, 8, 10 triangle.

For a formula that works on all rt. triangles:

if c is the hypotenuse and a and b are the other 2 sides

ab/a+b+c or

a+b-c/2

you should get the same answer