Geometry Custom Construction

So the other day, I thought of this problem which has pretty much wrecked my life.

Q : If the ratio of two sides, the included angle (let's say multiple of 15), and the perimeter of a triangle is known, is it possible to construct the triangle? (Construction meaning with no calculation, only ruler and compass)

I've thought of this for definitely over 12 hours, and I can't find a way, but maybe I missed something?

#Geometry

Easy Math Editor

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Comments

Well, if we assume the given perimeter and the ratio of 2 given sides to be $p$ and $r$ respectively, the 2 given sides of triangle come out to be $\frac{p}{1+r+\sqrt{1+r^{2}-2r\cos\theta}}$ and $\frac{rp}{1+r+\sqrt{1+r^{2}-2r\cos\theta}}$ where $\theta$ is the included angle. Now, you can take either of the calculated side as the base and then construct the other side at an angle $\theta$ with the base to construct the triangle. Ofcourse the construction of the sides depends on how "nice" the values of the parameters are given....

Aaghaz Mahajan - 1 year, 3 months ago

Thanks for the answer! However, I knew it could potentially be solved using trigonometry and my question was if it was possible without any calculations, only measuring the perimeter and angle. Of course, you could draw bisectors, or construct any angles you want (multiple of 15). Would it possible in such a case? Also could you explain how you got the formula for the lengths of the sides?

Akshaj Gopalakrishnan - 1 year, 3 months ago

Taking the ratio of sides to be $r$ we know that 2 sides of the triangle are $x$ and $rx$ with the angle $\theta$ included between them. Now to find the third side in terms of these variables, I used the Cosine Rule . Finally adding all the sides should give us $p$ and thus, after a bit of manipulation you arrive at the sides....

Aaghaz Mahajan - 1 year, 3 months ago

@Aaghaz Mahajan – Thank you for the quick reply! Do you think it is possible to do it without any calculation?

Akshaj Gopalakrishnan - 1 year, 3 months ago

@Akshaj Gopalakrishnan – You are welcome :)
Hmm I don't really think so......I mean you obviously would need to have the value of a side to construct a triangle uniquely. Or else all the possible triangles can satisfy the first two conditions ( i.e. ratio and the angle) but the perimeter poses a bound for the triangle to be unique....

Aaghaz Mahajan - 1 year, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$