Q:Consider $n+2$ distinct point from the circumference of a circle.If consecutive points along the circle are joined by line segments creating a polygon with $n+2$ sides then the sum of interior angle of the resulting polygon equals $180n$ degree.
My approach:It's trivial when $n=1$ forming a triangle.But how to prove it for Inductive step $n+1$?Making triangle give me a intuition but how to express it mathematically?
Can anyone give me some tips for this kind of mathematical induction.Any hint or solution will be appreciated.
Thanks in advanced.
[Math] proof by Mathematical induction related with geometry
induction
Best Answer
Using induction, you have already shown it's true for $n = 1$, i.e., for $3$ points, the sum is $180^\circ$.
Next, assume it's true for all $n \le m$ for some $m \ge 1$, i.e., the sum of the interior angles of $n + 2$ distinct points is $180n^\circ$. When going from a circle with $n + 2$ to $n + 3$ points, the next point must be between $2$ already existing nearest points on the circle. Draw a line from each of these $2$ nearest points to the new point (e.g., as in the circle on the right, a point between the end points of the horizontal line above the "$2$"). Since these are the nearest points, the new lines don't intersect any existing lines and, thus, they form a new triangle. Adding the angles of this new triangle, with a sum of $180^\circ$ to the existing ones gives a sum of interior angles of $180\left(n+1\right)^\circ$.
Thus, if it's true for $n$, it must be true for $n + 1$. This completes the induction process to prove it's true that the sum of interior angles of $n + 2$ distinct points on a circle is $180n$ degrees for all $n \ge 1$.
FYI, here is an alternate method you can use without induction. For $n + 2$ points, draw the lines from the center of the circle to each of these points. This will form $n + 2$ triangles. Thus, the sum of the angles of all these triangles is $180\left(n + 2\right)$ degrees. However, this includes the angles at the center of the circle, which together add to $360$ degrees. Thus, subtracting this gives that the sum of internal angles is $180n$ degrees.