## Monday, October 1, 2018

### The Largest Rectangle Inscribed in Any Triangle

From Mathcounts Mini : Maximum area of inscribed rectangles and triangles

$$\Delta EHI\sim\Delta EFG$$ $$\rightarrow$$ $$\dfrac {a} {c}=\dfrac {d-b} {d}$$$$\rightarrow$$ $$a=\dfrac {c\left( d-b\right) } {d}=\dfrac {-c\left( b-d\right) } {d}$$

We are going to find out what the largest area of a rectangle is with the side length a and b.
It can be shown that by substituting the side length "a" with the previous equation + completing the square that the largest area is half of the area of the triangle the rectangle is embedded.

$$a\times b=\dfrac {-c\left( b-d\right) \times b} {d}=\dfrac {-c\left( b^{2}-bd\right)} {d}= \dfrac {-c\left( b-\dfrac {1} {2}d\right) ^{2}+\dfrac {1} {4}dc} {d}$$.

From there, you know that when $$b= \dfrac {1} {2}d$$, it will give you the largest area, which is $$\dfrac {1} {4}dc$$.

$$a=\dfrac {-c\left( b-d\right) } {d}= \dfrac {-c\left( \dfrac {1} {2}d-d\right) } {d}=\dfrac {c\left( d-\dfrac {1} {2}d\right) } {d}=\dfrac {1} {2}c$$.

Thus, the maximum rectangle area occurs when the midpoints of two of the sides of the triangle were joined to make a side of the rectangle and its area is thus 50% or half of the area of the triangle or 1/4 of the base times height.

Proof without words from Mr. Rusczyk

Try using different types of triangles to experiment and see for yourself.
Paper folding is fun !!!!!
It's very cool :D

#### 1 comment:

1. A nice proof to an interesting problem! Thanks for posting.