****Spoiler Alert: This post contains the full solution****
First let's make sure we have the right picture.

So

and

are sitting on the

-axis, while

is somewhere to the right of the

-axis at height 1. For this dataset, leave-one-out validation entails fitting our model to two of the points, then testing the fit on the third. Let's start with the constant model,

. When we fit this model on two data points,

will simply be the average of the

-coordinates of the two points.

- Leaving out, we find . The error is .
- Leaving out, we also find . Again, .
- Finally, leaving out, . The error is .

The overall cross-validation error is the average of the three individual errors,

, as you can verify. Looking ahead, we would like to find the value of

that makes

.

Let's turn to the linear model,

. The easy case is when

is left out. The resulting fitted line is simply

and the error is

.

Things get more complicated when

is left out. We need to find the equation of the line through

and

. Using slope-intercept form and rearranging, you can check that the fitted line has slope equal to its intercept,

. The error on

is

.

A similar derivation yields

.

Putting it all together gives us

. If we set this equal to 1/2 (the error from the constant model), we have a quadratic equation in one unknown, which we can solve using the quadratic formula (alternatively, dumping the whole equation into WolframAlpha gives you the roots directly).

Hope that helped!