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

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## Solution

The angle α is 15°.

## Poem

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

## 3 replies on “The spotlight”

a=15 graden

Heel goed Marian, hoe heb je dat gevonden?

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.