MHB Circumcircles Find a formula relating R,R_1,R_2.

  • Thread starter Thread starter mrtwhs
  • Start date Start date
  • Tags Tags
    Formula
AI Thread Summary
In the discussion, a right triangle ABC with legs BC = a, AC = b, and hypotenuse AB = c is analyzed to derive a relationship between its circumradius R and the circumradii R1 and R2 of two isosceles triangles formed by joining two copies of triangle ABC. The isosceles triangle formed with a as the common side has sides c, c, and 2b, leading to circumradius R1. Conversely, the isosceles triangle formed with b as the common side has sides c, c, and 2a, resulting in circumradius R2. The participants are focused on finding a formula that connects R, R1, and R2 based on these configurations. The discussion emphasizes geometric relationships and circumradius calculations in right triangles.
mrtwhs
Messages
47
Reaction score
0
Let $$\triangle ABC$$ be a right triangle with right angle at $$C$$. Suppose this right triangle has legs $$BC=a$$, $$AC=b$$, hypotenuse $$AB=C$$, and circumradius $$R$$. Two copies of this triangle can be joined to form an isosceles triangle in two ways. With $$a$$ as a common side, you can form an isosceles triangle with sides $$c,c,2b$$ and circumradius $$R_1$$. With $$b$$ as a common side, you can form an isosceles triangle with sides $$c,c,2a$$ and circumradius $$R_2$$. Find a formula relating $$R,R_1,R_2$$.
 
Mathematics news on Phys.org
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:
$$c^2=a^2+b^2=\frac{c^4}{4R_1^2}+\frac{c^4}{4R_2^2}$$
$\implies\ \dfrac1{c^2}=\dfrac1{4R_1^2}+\dfrac1{4R_2^2}$

$\implies\ \boxed{\dfrac1{R^2}\ =\ \dfrac1{R_1^2}+\dfrac1{R_2^2}}$.
 
Last edited:
Olinguito said:

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


Nice! Pretty much the same way I solved it.
 
Last edited by a moderator:
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top