The spotlight

A right triangle with two cevians. What’s the angle α?

Scroll down for a solution to this problem.


The angle α is 15°.

Solution by Matthew Arcus.


Today we live in Covidie
Which puts the spotlight
On world’s economy
News and fake news all day
That does not help the virus to go away
A viral world, what a dismay
The virus will destroy and stay
No hugs, no hand shake
No kissing
All news by mail
And all our wishes just fail


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3 replies on “The spotlight”

I did this the hard way. 🙂

Start with tan(4a)-tan(3a) = tan(3a)-tan(a).
Set x = tan(a), and then have
tan(4a)= (4x-4x^3)/(1-6x^2+x^4)
tan(3a) = (3x-x^3)/(1-3x^2).
Need to solve:
tan(4a)+tan(a) = 2*tan(3a),
which translates to:
(4x-4x^3)/(1-6x^2+x^4) + x == 2*(3x-x^3)/(1-3x^2).
Divide by common factor x.
Then replace x^2 by t.
Clear denominators, and get
t^3 – 13*t^2 -13*t + 1 == 0
which factors (t^2 – 14*t + 1)*(t + 1)
The root t=7-sqrt(3) gives x = 2-sqrt(3), which is the tangent of 15 degrees.

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