By: Fhonda Danley & September Matteson

`EMT 469 - WTR 96`

` Starting with the red triangle, ABC, we constructed three equilateral
triangles on the exterior of ABC. Next, we found the centers of each of these
equilateral triangles and joined these centers to form the blue triangle. This
triangle can also be known as Napoleon's triangle.`

`
`

`Our question now is whether or not this blue triangle is indeed Napoleon's
triangle. One easy way to investigate this would be to measure the sides of
the blue triangle and see if they are congruent, however; this would only give
us a hypothesis and not an actual proof. So, let's try this and see if we get
this result for a hypothesis.`

`
`

`As we can see, from the picture, all of the sides are indeed congruent,
giving us 5.07 cm for the length. Now we can give a hypothesis.`

__Hypothesis__:**Given any type of triangle ABC, whenever**
**we construct three equilateral triangles on each side of the original, if we
join the centers of these triangles our newly formed triangle will be a
Napoleon's triangle.**

`Now, let's investigate this using different types of triangles as our
triangle ABC.`

`
`

`For this picture, above, we can see what happens when our triangle ABC is
an equilateral triangle. Our result is the blue triangle where once again all
of the sides are congruent. Now let's see if we get the same results when our
triangle ABC is isosceles.`

`
`

`From this, we see what happens when our ABC is isosceles(its measurements
are shown on the left side of the picture). We see that the sides of our blue
triangle are indeed congruent once again. Let's investigate what happens when
the triangle is right triangle.`

`
`

`As we can see, once again our newly formed triangle, yellow triangle, has
all lengths congruent. Therefore, by definition of Napoleon's triangle, it is
indeed.`

`Now, I think that we can safely say that by way of our hypothesis, we have
proven what we set out to prove. That yes indeed whenever you have any given
triangle where three equilateral triangles are formed on its exterior, when the
centers of these triangles (equilateral) are joined, the resulting triangle
will be a Napoleon's triangle (where all of its sides are congruent).`