Triangle, Bisectors and Radius of Circumcircle

Radius of a Circumcircle

Given a triangle with vertices A, B and C, find a formula for the radius R of the circumcircle.
Let A, B and C be the the vertices of a triangle. Let L1 be the perpendicular bisector (line through the midpoint of the opposite side to vertex A and perpendicular to this side) of segment BC, L2 the perpendicular bisector of AC and L3 the perpendicular bisector of AB. These 3 lines intersect at one point O. This point is the center (circumcenter) of a circle called circumcircle passing through the vertices A, B and C of the triangle.

perpedicular bisectors, circumcircle in triangle

Angle BOC is twice angle BAC since they intercept the same chord BC but BOC is a central angle Euclid's Elements, Book III, Proposition 20 and central and inscribed angles.
In triangle BOC, BM = MC. Which leads to angle BOM = angle MOC which leads to angle BOM = angle MOC = angle A.
In triangle BOM,

sin(BOM) = BM / OB

But
BM =BC/2
Which leads to:
sin(BOM) = BC / (2 OB)

angle BOM = angle A and OB is the radius R of the circle.
Hence
sin(A) = BC / (2R)
or
2R = BC / sin(A)
and now using the sine law, we have
2R = BC / sin(A) = AC / sin(B) = AB / sin(C)

Interactive Tutorial on Circumcircles

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1 - click on the button above "click here to start" and MAXIMIZE the window obtained.

2 - You can change the position of a vertex A, B or C the triangle by dragging it. The perpendicular bisectors are shown in in red. Drag the center of the circle (in black) and use the slider top of left panel to adjust the radius of the circle untill all three vertices are on the circle. This is the circumcircle.

3 - Repeat step 2 untill for different positions of A, B and C.

4 - Change the position of A and/or B and C to make angle A equal to 90 degrees. Values of angles and side lengths are given. The point of intersection of the bisectors (circumcenter) is on one side of the triangle. Explain.

5 - The radius R of the circumcircle is given by

R = BC/(2*sin(A)) = AC/(2*sin(b)) = BA /(2*sin(C))

Change the positions of A, B and C and use the values of the lengths of AC, BA and BC and angles A, B and C to find radius R. Compare this value to the radius given by slider (top left).

6 - Repeat step 5 as many times as you wish.

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