Triangle drawn entirely within a circle?

Question

Show that the sum of the sides of the triangle is less than the circumpherence of the circle.

The triangel has sides length a, b, c

The circle is radius r

So the question is: Show that a+b+c < 2 pi r

The points of the triangle do not touch the circle.

Answer 1

The maximum possible length of any straight line in a circle is the diameter ie 2r.

The circumference of the circle is Pi x 2r.

Therefore the circumference must always be at least 3.14 times the length of the longest line in the triangle. As I triangle only has three sides it is impossible for the sum to be more than the circumference.

QED

Answer 2

Let C = 2pi r, the circumference of a circle. As an equilateral triangle has 60 deg internal angles at each apex, the arc length L subtended by one corner angle is L = 2pi r/3 .

The length of chord associated with the arc is a = 2r sin(30) = 1/2 2r = r. Thus L/a = (2pi r/3)//(r) = (2 pi/3)/(1) = 2/3 pi > 1.0; so L > a that is, the subtended arc on the circle is larger than any one of the three sides since the sides are equal in length.

We can write C = 3L > 3a = a + b + c; so that 2 pi r > a + b + c QED.

Answer 3

For each edge of the triangle there is a curve, part of the circle. The edge is a straight line - its the shortest line between two points. But the section of the circle between the same two points is longer - in all 3 cases. So adding all the 3 sides together must be less than adding the whole circumference sections together.

Answer 4

For each edge of the triangle there is a curve, part of the circle. The edge is a straight line - its the shortest line between two points. But the section of the circle between the same two points is longer - in all 3 cases. So adding all the 3 sides together must be less than adding the whole circumference sections together.

Answer 5

This is a rather crude proof but good nonetheless

Consider the theoretical maximum length of one side of the triangle... it would be equal to the length of the largest side that can be drawn in a circle. And that is its diameter. So the maximum length of one side can be 2r. Similarly, assuming the worst case (not possible), consider all sides to be equal to this side. Then you have

a + b + c <=3*2*r

Replace 3 with 3.141592... (pi), and remove the =

a + b + c < 2 * pi * r
Q.E.D

Answer 6

All the three sides are chords of the ARC between the vertices of the triangle. The sum of the Arc lenghts is circumferance & sum of corrosponding Chord lengths is the perimeter of the Triangle.

In a circle Chord length is always less than the Arc length & hence the answer

Answer 7

Graphical Solution: Draw a line connecting Islands A and B, bisect this line and draw a perpendicular line on the midpoint Draw a second line connecting B and C, bisect this line and draw a perpendicular line on the midpoint. Extend the perpendicular line between islands A & B, same with the perpendicular line between islands B & C. There will be a meeting point between these lines. The point these lines will intersect. The intersection point is the center of the circle you are looking for. Now with the aid of a drawing compass, put the fix point on the center of the circle you have drawn and the other end reach one of the islands and draw the circle. You will see that the three islands have intersected on the perimeter of the circle you have made. You should draw this on graphing paper. For the area, this is the formula, Area of circle = PI r^2 where PI = 3.1416 and the answer will be in km^2

Answer 8

One of the lines of the triangle must be the diameter (call it AC). so circumfrence minus pi x diameter = 2.14 diameter. SinceAC is the hypotenuse, AB and BC are shorter. (get it ? no?)

2.14 diameter minus AB leaves a remainder likewise with BC. We come to the conclusion a+b+c<2pi r