Let \(\bigtriangleup ABC\) be a triangle and \((\bigtriangleup ABC)=AB+BC+CA\). How is this possible and what's the value of \(AB,BC\) and \(CA?\)

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TopNewestIt is possible for right triangles with side lengths: \((5,12,13)\) and \((6,8,10)\).

Actually they are the only right triangles that satisfy this condition (perimeter equals area).

For its proof, see here.

Hope this helps!

EDIT: By the way, I assumed that you were talking about integer side-lengths and area. – Mursalin Habib · 4 years ago

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How would you deal with this for a general triangle?

The assumption of integer side-lenghts (and hence area) is reasonable. How would your interpretation change if the question was for any real side lengths? – Calvin Lin Staff · 4 years ago

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– Jimmy Kariznov · 4 years ago

For every set of angles \((A, B, C)\) that sum to \(180^{\circ}\), set \(R = \dfrac{\sin A + \sin B + \sin C}{\sin A \sin B \sin C}\). Then, the triangle with sides \(BC = 2R\sin A\), \(CA = 2R\sin B\), \(AB = 2R\sin C\) has area and perimeter equal to \(\dfrac{2(\sin A + \sin B + \sin C)^2}{\sin A \sin B \sin C}\). This gives us uncountably many triangles.Log in to reply

For a more direct approach, if a triangle has area \(A\) and perimeter \(P\), then if you scale the triangle by \( \frac{P}{A} \), it will have area \( A \left( \frac{P}{A} \right)^2 = \frac{P^2}{A} \) and perimeter \( P \frac{P}{A} = \frac{P^2} { A} \).

The \(R\) that you calculated reflects the above scaling. – Calvin Lin Staff · 4 years ago

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It turns out there are \(5\) triangles with integer side lengths and equal perimeter and area.

They are triangles with side lengths:

\((5, 12, 13)\)

\((6, 8, 10)\)

\((9,10,17)\)

\((7, 15, 20)\)

And \((6, 25, 29)\).

I started off by letting the side-lengths be \(a\), \(b\) and \(c\) and used Heron's formula for the area. And then it turned into a number theory problem.

A good solution to that can be found here and here.

I also learned that such triangles are called perfect triangles.

As for the side lengths being any real (don't you mean positive real?) numbers, Jimmy has shown that there are infinitely many such triangles. – Mursalin Habib · 4 years ago

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Here's a problem though: the units are not the same for both sides. It's length squared for area while it's just length for the sum of the sides. – Xuming Liang · 4 years ago

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– Jimmy Kariznov · 4 years ago

In that case, let \(u\) be a unit distance, and let \(BC = au\), \(CA = bu\), \(AB = cu\). Find all triples \((a,b,c)\) of positive integers such that \([\Delta ABC] = (AB+BC+CA) \cdot u\).Log in to reply