# When two triangles have the same heights?

Let $$\Delta ABC$$ and $$\Delta EFD$$ be triangles with sides $$a$$, $$b$$, $$c$$ and $$d$$, $$e$$, $$f$$. All sides have different lengths.

I am looking for a functional condition for sides when triangles can have the same heights $$h_E=h_B$$.

My attempt.

I have found the three cases when two triangles can have the same hights.

Case 1. Acute triangles

Case 2. Acute triangle and obtuse triangle

Case 3. Obtuse triangles

• How do you define the height of a triangle in general?, for example if you rotate some of your triangles the "height" may not visioned as height any more. This may be useful:books.google.com.eg/… Sep 7, 2019 at 1:08
• @NoChance, I can rotate both triangles.
– Nick
Sep 8, 2019 at 0:25

If you only want to use the edge lengths, use Heron's formula to get $$A_1 = \frac{1}{4} \sqrt{(a + b + c)(-a + b + c)(a - b + c)(a + b - c)}$$ $$A_2 = \frac{1}{4}\sqrt{(d + e + f)(-d + e + f)(d - e + f)(d + e - f)}$$
So two triangles have the same height if $$\frac{\sqrt{(a + b + c)(-a + b + c)(a - b + c)(a + b - c)}}{b} = \frac{\sqrt{(d + e + f)(-d + e + f)(d - e + f)(d + e - f)}}{e}$$
Minimum distance between two parallel lines is the same, is the altitude $$h$$ of all triangles no matter where you choose the base and shift the base segment length AC or DF along this line AF and no matter where the apex B or E lies.