Descartes' Circle Formula is a relation held between four mutually tangent circles. , the opposite of the reciprocal of its radius.

Descarte's rule of signs is a method used to determine the number of positive and negative roots of a polynomial. The rule gives an upper bound on the number ...

Art of Problem Solving AoPS Online Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online Books for Grades 5-12 Online Courses

The problem asks us to find the sum of the reciprocals of the square roots of the radii in each circle in this collection. Problem · Solution 1 · Solution 2 (Pythagorean...

14 янв. 2018 г. · I was looking through the 2017 AMC 12A and found that I could use Descartes' Circle Theorem on #16. It's probably more simpler than using ...

Solution 2. Note that there are no sign changes (all coefficients of terms are positive), so by Descartes' rule of signs, there are no positive real roots ...

Three circles of radius $1$ are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle?

LaTeX for Descartes' Circle Theorem Handout as well as rendered PDF. First published on AoPS here: https://artofproblemsolving.com/community/c1967976h2700478

Using Descartes' theorem, we get $k_4=\frac15+\frac15+\frac18+2\sqrt{\frac . Thus, $r_4=\frac{1}{k_4}=\frac{40 , and the answer is $\boxed{017}$. Video ...

This problem can be very easily solved using Descartes' Circle Theorem. It states that if we have 4 circles that are all tangent with each other,