Equilateral Triangle -- Elegant Solution

Geometry Level 3

We've an equilateral triangle PQR with a point S inside such that <PQS=54° & <PRS=48°. The task on hand is to determine <RPS preferably w/o using trigonometry.

The answer is 18.

2 solutions

Nibedan Mukherjee
May 31, 2020

Though I solved by Ceva's Trig approach ... still trying to figure out the euclidean way... ;P

Looks promising!

ajit athle - 1 year ago

means lot sir...! , Sir I'm trying to solve it.. by alpha drift transformation, and fagnano's decomposition..

nibedan mukherjee - 1 year ago

Ajit Athle
May 28, 2020

One may use Ceva's Theorem in trigonometric form to arrive at the solution which is x=18°. [sin(x)/sin(60-x)][sin(54)/sin(6)][sin(12)/sin(48)]=1. However, what I am looking for is an elegant solution which, I hope, someone will provide.

Nice question. I like the emphasis on an elegant solution. I've not managed to find one yet (I also used trig; a couple of applications of the sine rule which amount to the same approach as yours), but I'm interested in whether there is an elegant solution.

Just some initial thoughts:

One thing to check might be what happens if you change the values of the two given angles; do you still get a "nice" answer for $x$ (where "nice" might mean, for example, an integer number of degrees for integer inputs)? If so, there might be an angle-chasing solution.

If not, what's special about the values $18$ , $48$ and $54$ ? What other triples of integers work? Is there a geometric approach using reflected/rotated copies of the original triangle?

Chris Lewis - 1 year ago

The beauty of these three numbers is that their sum is $120$ , the vertex angle of an isosceles triangle with each base angle $30\degree$ , half of the angles of an equilateral triangle. I tried in this direction but in vain. I'm weak in Mathematics.

A Former Brilliant Member - 1 year ago

Equilateral Triangle -- Elegant Solution

The answer is 18.

2 solutions

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