Circumcircles Find a formula relating R,R_1,R_2.

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In summary, the conversation discusses a right triangle with legs BC=a, AC=b, and hypotenuse AB=C, as well as its circumradius R. It is noted that two copies of this triangle can be joined to form an isosceles triangle in two ways, with sides c,c,2b and circumradius R_1 when a is the common side, and with sides c,c,2a and circumradius R_2 when b is the common side. The conversation concludes with the task of finding a formula that relates R, R_1, and R_2.
  • #1
mrtwhs
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Let \(\displaystyle \triangle ABC\) be a right triangle with right angle at \(\displaystyle C\). Suppose this right triangle has legs \(\displaystyle BC=a\), \(\displaystyle AC=b\), hypotenuse \(\displaystyle AB=C\), and circumradius \(\displaystyle R\). Two copies of this triangle can be joined to form an isosceles triangle in two ways. With \(\displaystyle a\) as a common side, you can form an isosceles triangle with sides \(\displaystyle c,c,2b\) and circumradius \(\displaystyle R_1\). With \(\displaystyle b\) as a common side, you can form an isosceles triangle with sides \(\displaystyle c,c,2a\) and circumradius \(\displaystyle R_2\). Find a formula relating \(\displaystyle R,R_1,R_2\).
 
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  • #2
My attempt:

For the first circle, we clearly have
$$R=\frac c2$$
as the circumcentre is the midpoint of the hypotenuse $\mathrm{AB}$.

For the second circle, let $\mathrm D$ be the point on $\mathrm{BC}$ extended so that $\angle\,\mathrm{DAB}$ is a right angle. Then $\triangle\mathrm{DCA}$ is similar to $\triangle\mathrm{ACB}$ and so $|\mathrm{DC}|=\dfrac{b^2}a$. The circumcentre is the midpoint of $\mathrm{DB}$ and so
$$R_1=\frac12\left(\frac{b^2}a+a\right)=\frac{c^2}{2a}.$$

By symmetry,
$$R_2=\frac{c^2}{2b}.$$

Hence:
$$c^2=a^2+b^2=\frac{c^4}{4R_1^2}+\frac{c^4}{4R_2^2}$$
$\implies\ \dfrac1{c^2}=\dfrac1{4R_1^2}+\dfrac1{4R_2^2}$

$\implies\ \boxed{\dfrac1{R^2}\ =\ \dfrac1{R_1^2}+\dfrac1{R_2^2}}$.
 
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  • #3
Olinguito said:

$\boxed{\dfrac1{R^2}\ =\ \dfrac1{R_1^2}+\dfrac1{R_2^2}}$.

Nice! Pretty much the same way I solved it.
 
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FAQ: Circumcircles Find a formula relating R,R_1,R_2.

What is a circumcircle?

A circumcircle is the circle that passes through all the vertices of a given triangle.

What are R, R1, and R2 in the formula?

R is the radius of the circumcircle, and R1 and R2 are the radii of the circles inscribed in the triangle formed by the three sides of the circumcircle.

What is the formula for finding the radius of a circumcircle?

The formula for finding the radius of a circumcircle is R = (abc)/(4√(s(s-a)(s-b)(s-c))), where a, b, and c are the side lengths of the triangle and s is the semi-perimeter (s = (a + b + c)/2).

How do I use the circumcircle formula to find the radius?

To use the formula, you need to know the side lengths of the triangle. Plug those values into the formula and solve for R.

Can the circumcircle formula be used for any type of triangle?

Yes, the circumcircle formula can be used for any type of triangle, including equilateral, isosceles, and scalene triangles.

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