Use vectors to demonstrate the following fact On a circle an

Use vectors to demonstrate the following fact: On a circle any two diametrically opposed points along with an arbitrary third point (on the circle) form a right triangle.
Hint: Assume without loss of generality that the circle is centered at the origin and let v, v and w denote the three points in question. Show that the vector connecting w to v is orthogonal to the vector connecting w to v.

Solution

Take center of circle to be origin

The two diametrically opposite vectors be:v and -v

The other arbitrary third point on the circle is representing by vector w ,starting from origin and ending on the circle

So vector from w to -v is:w-v

Vector from w to v is :w+v

(w-v).(w+v)=w.w-v.v=|w|^2-|v|^2

Note that both w and v start on origin and end on the circle and hence their magnitudes are equal and equal to radius of circle

So, (w-v).(w+v)=|w|^2-|v|^2=0

Hence the three points form a right triangle.