by Oktay Mercimek
I assume I ask these question to my audience and audience is a high school class
Q1: What is the circumcenter and circumcircle ?
A2: Circumcenter is the center of circumcircle that a triangle's vertexes lie on it.
Q2: How can we find circumcenter for given a triangle ABC ?
A2: At first we should think about the definition of a circle.
As we know circle is set of points that are equal distance (r) to a fixed point.
So we are looking a circle that incident with points A, B and C.
Then center of circle must be equal distance from point A, B and C.
Lets find points that equal distance from A and B.
Q3: How can we find points that equal distance to A and B.
A3: Midpoint of AB comes to mind and first. Let's think an isosceles triangle ABP that
AB is the base of triangle.
If ABP is a isosceles triangle then a perpendicular line l to Segment AB trough P is the bisector for segment AB.
Moreover every point on l is equal distance to A and B.
We can apply same thought to AC. Let's m to be perpendicular bisector to AC.
So point T is equal distance from A, B and C.
|AT| = |BT| . Because T lies on l.
and |AT| = |TC| because T also lies on m.
Then |AT| = |BT| = |TC|
Since |BT| = |TC| , triangle BTC s a isosceles triangle and segment BC is the base of triangle.
If we draw a perpendicular line n to BC that incident with P, then this line has to be a perpendicular bisector to segment BC, and that is how the three perpendicular bisector of the sides of a triangle have to be concurrent.
Let's turn to our question 2 . Since point T is equal distance from A, B and C ,we can define a circle that T is the center.
And this is the circumcenter for triangle ABC. Click HERE for GSP demonstration.