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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.