Try this
Drag the orange dot on the triangle below. The triangle will have a fixed
perimeter, but the
area will vary.

A common error is to assume that a triangle that has a fixed
perimeter
must also have a fixed
area.
This is definitely *not* the case as can be seen from the figure above. As you drag the orange point A,
the triangle will maintain a fixed perimeter. But as you can see, the area varies quite a bit.

When A is half way between B and C, the area is at a maximum. As you drag it around towards one side you can see the area decreasing, both in the formula at the top and by noticing that fewer and fewer squares can fit inside it. Eventually, when A is in line with B and C, the area is zero.

The area is at a maximum when the triangle is isosceles. That is, when both sides have the same length. Carefully adjust A above to create an isosceles triangle and note the area is the greatest when AC and AB are both the same length (9.0)

Why is this? The definition of an ellipse is

"A line forming a closed loop, where the sum of the distances from two points (foci) to every point on the line is constant"

Points B and C form the two foci. Since the distance from B to C is fixed, and the perimeter is fixed, then the sum of the distances AB and AC are
constant - the condition required to form an ellipse.
The string experiment described above is actually a practical way to draw an ellipse. See Drawing an ellipse using string and 2 pins. For more on ellipses, see also Definition of an ellipse.

- Triangle definition
- Hypotenuse
- Triangle interior angles
- Triangle exterior angles
- Triangle exterior angle theorem
- Pythagorean Theorem
- Proof of the Pythagorean Theorem
- Pythagorean triples
- Triangle circumcircle
- Triangle incircle
- Triangle medians
- Triangle altitudes
- Midsegment of a triangle
- Triangle inequality
- Side / angle relationship

- Perimeter of a triangle
- Area of a triangle
- Heron's formula
- Area of an equilateral triangle
- Area by the "side angle side" method
- Area of a triangle with fixed perimeter

- Right triangle
- Isosceles triangle
- Scalene triangle
- Equilateral triangle
- Equiangular triangle
- Obtuse triangle
- Acute triangle
- 3-4-5 triangle
- 30-60-90 triangle
- 45-45-90 triangle

- Incenter of a triangle
- Circumcenter of a triangle
- Centroid of a triangle
- Orthocenter of a triangle
- Euler line

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