While attempting to solve a challenging math question, I stumbled upon a identity that I have never seen before. It was definitely not the intended solution, but while playing around with it, I was able to come up with formula regarding the angles of a triangle.
I tried to find if someone else has come up with this formula before, but from my experience, I wasn’t able to locate any.
I thought I would just post it here as a reference since it is a property that I’m proud of finding, and if I am really the first one, it would act as a proof that I came up with it. I found this property on July 24th 2020 and I am posting now after a bit of research and clearing things up.
The property is essentially about “squeezing” a triangle in a direction parallel to the altitude and how that affects the angles.
Referring to the image above,
tan(a+b)/tan(a) = tan(c+d)/tan(c)
where m+n and m are altitudes of the original and squeezed triangles respectively.
This is the formula.
Here is the proof.
tan(a+b) = (n+m)/x
tan(a) = m/x
tan(a+b)/tan(a) = n+m/m
tan(c+d) = (n+m)/y
tan(c) = m/y
tan(c+d)/tan(c) = (n+m)/m
∴ tan(a+b)/tan(a) = tan(c+d)/tan(c)
This was a very interesting property to find, especially since I have never been taught anything similar! It would be really surprising if no one else came up with it before.