The Inscribed Angle Property: (also known as the Central Angle Theorem) The measure of an angle inscribed in a circle is half the measure of the arc it intercepts. It follows that all inscribed angles that intercept a given arc have equal measure.
fig 1: center of circle is on BC
fig 2: center of circle not inside angle
fig 3: center of circle inside angle
Figures 1, 2, and 3 show three cases, which differ in the matter of whether the center of the circle is in the interior of the inscribed angle. In every case, the measure of angle ABC is exactly half the measure of AOC.
In figure 1, this fact is practically self-evident. An exterior angle of a triangle is the sum of the other two interior angles, so the measure of AOC is equal to the sum of the measures of OAB and OBA. OAB is an isosceles triangle (because radii OB and OA are equal) so the measures of angles OAB and OBA are equal. So the measure of angle AOC is twice the measure of angle ABC.
To see this in figure 2, draw one more line, a diameter, from B through O meeting the circle at point D:
|Now as we have seen, angle COD is twice angle CBD. By the same reasoning, angle AOD is twice angle ABD. The measure of angle AOC is the difference of the measures of AOD and COD, and the measure so the measure of angle AOC is twice the difference of the measures of ABD and CBD, and that difference is the measure of ABC, so the measure of AOC is twice the measure of ABC.|
Finally, let's look at the case in figure 3. We will add the diameter, just as we did in the previous case.
|Since BD is a diameter, just as in the previous case, DOC is twice DBC, and AOD is twice ABD. So AOC (the sum of AOD and DOC) is twice ABC (the sum of ABD and DBC).|
Summary of geometrical theorems
The Intersecting Chords Theorem is a generalization of the central angle theorem which was presented above.
The other Intersecting Chords Theorem says the products of the two segments of chords cut by their point of intersection are equal.
The proofs of these theorems use the Inscribed Angle property of circles:
Law of Sines - Given triangle ABC with opposite sides a, b, and c, a/(sin A) = b/(sin B) = c/(sin C) = the diameter of the circumscribed circle.
Circumscribed Circle - The radius of a circle circumscribed around a triangle is R = abc/(4K), where K is the area of the triangle.
Ptolemy's Theorem uses the facts presented here.
The Median and Altitude of a Right Triangle are reflections about the Right Angle Bisector
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