Question

abc is a right-angled triangle. Right-angled triangled at B and O is the mid-point of AC. Show that O is equidistant from points A,B and C.

Answer 1

Question

As we know all triangles are cyclic, i.e. every triangle has a circumscribed circle. And in our case the triangle $ABC$ has a right angle $\angle B = 90{}^\circ$. So, we can say that chord $AC$ is a diameter of the circumscribed circle, because angle $\angle B = 90{}^\circ$. In this case distance from point $O$, which is the midpoint of the chord $AC$ to every vertex of the triangle $ABC$ are equal. In other words $OA = AB = OC = R$, where $R$ is the radius of the circumscribed circle. Proved.

**Answer**: Proved: $OA = AB = OC = R$, where $R$ is the radius of the circumscribed circle.