With that done, we then thought to inaugurate an irregular series on this blog, where we feature our own problems that have been published elsewhere.
Welcome to the first edition of such a series!
The three roots of the cubic are the slopes of the sides of a triangle. Find the slope of its Euler line. Details here.
Among all the proposed problems for that edition, the above was the easiest, as seen from the nature of the question and the number of solutions below.
Here’s a screenshot of the editor’s preferred solution:
A few things to point out in addition to the solution above.
We already saw that the slope of the Euler line of this particular triangle is from the screenshot above. We show that one side of the triangle has slope . Indeed, is a root of the cubic equation , since
So the Euler line is perpendicular to a side of the triangle.
Previously we found as one of the roots of the cubic equation , and so is an associated factor of the cubic. By polynomial division we obtain as the second factor, and so the other two roots can be obtained from the quadratic equation :
where is the golden ratio.
The roots are . They form a geometric progression because
This follows from the fact that the slopes of the sides of the triangle form a geometric progression with as the geometric mean. (Same result if the geometric mean was .)
Since the slopes of the sides form a geometric progression with a common ratio , it follows that the slopes of the medians also form a geometric progression.
Let be a real number. Then the following statements are equivalent:
- the cubic equation has (three) real roots
- or .
- (Symmetric equations) Consider the cubic equation .
- Show that is always a solution.
- Find a condition on for which is the only real solution.
- (Social engagement) The problem below was posted on a Facebook group recently: