Another purportedly hard geometry puzzle

9 min readNov 7, 2023

Part 1 of 3: Formulating the Problem

A circle is nestled between three curves on the positive x y plane. First, the concave edge of a circle of radius one centred at the origin. Second, the convex edge of the curve y equals the square root of x. Third, the y axis. What is the radius of the circle? — Figure 1: A purportedly hard geometry puzzle

One can rotate Figure 1 anticlockwise by 90° and flip it horizontally without changing the radius of the smaller circle. The hope is that this will make the problem easier to solve. This is Figure 3 below.

Notice how √x has become x². That’s because rotating switches the x and y for each other. It problem suddenly looks much better! Now, let’s name some points of interest. These are shown in Figure 4.

These points of interest are the following: the centre (a, b) and radius (r) of our circle (the smaller circle), and the three points of tangency ((a1, a2), (b1, b2) and (c1, c2)) of the three curves that form tangents with our circle respectively.

Curve one, x squared plus y squared equals one. Curve two, y equals x squared. Curve three, y equals zero. The radius of our circle of interest, r. The centre of our circle of interest, a comma b. The point of tangency of curve two, y equals x squared, with our circle, a one comma a two. The point of tangency of curve three, y equals zero, with our circle, b one comma b 2. The point of tangency of curve one, x squared plus y squared equals one, with our circle, c one comma c 2. — Figure 4: Marking the points of interest and specifying the curves’ equations

There are nine unknowns in Figure 4, so one needs nine independent equations to find all the unknowns. A circle is uniquely determined by three variables: r, a, b, i.e., its radius and the x- and y- coordinates of its centre.

Another purportedly hard geometry puzzle

Part 1 of 3: Formulating the Problem

Written by Dheeraj Dhobley