Circle Theorems

Revision as of 16:58, 29 April 2019 by Zhang2018 (talk | contribs) (synopsis)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Below is a brief introduction to the $10$ most fundamental circle theorems.

Theorem $1$: If a line is drawn from the center of a circle to the midpoint of a chord, the lines are perpendicular. Vice versa, if a line is drawn from the center of a circle and is perpendicular to the chord, the point of the intersection is the midpoint of the chord.

[asy]draw(circle((0,0),5)); draw((0,0)--(0,5)); draw((-3,4)--(3,4)); draw(rightanglemark((0,0),(0,4),(3,4))); [/asy]

Theorem $2$: If $2$ radii are drawn from the center of the circle and the points where the $2$ radii lines intersect the circle meet at another point, the interior degree measure created by the radii is $2$ times the measure of the interior degree measure of the angle created by the chords. This theorem is rather visual, so refer to the pictures below. Remember the shapes the lines create must be an arrow head or a double triangle, as shown in pictures $1$ and $2$, respectively.



Arrow Head [asy]draw(circle((0,0),5)); draw((-3.15,-3.883)--(-0.99, 4.901)); draw((2.59, -4.277)--(-0.99, 4.901)); draw((-3.15, -3.883)--(0,0)); draw((2.59, -4.277)--(0,0)); dot((-3.15,-3.883)); label("A",(-3.15,-3.883),S); dot((2.59,-4.277)); label("B",(2.59,-4.277),S); dot((-0.99,4.901)); label("C",(-0.99,4.90 1),N); dot((0,0)); label("D",(0,0),N); markscalefactor=0.08; draw(anglemark((-3.15,-3.883),(0,0),(2.59,-4.277))); label("${2x}^\circ$",anglemark((-3.15,-3.883),(0,0),(2.59,-4.277)),S,green); draw(anglemark((-3.15,-3.883),(-0.99,4.901),(2.59,-4.277))); label("${x}^\circ$",anglemark((-3.15,-3.883),(-0.99,4.901),(2.59,-4.277)),SE,green); [/asy]



Double Triangle

[asy]draw(circle((0,0),5)); draw((-3.15,-3.883)--(4.913,-0.929)); draw((2.59, -4.277)--(4.913,-0.929)); draw((-3.15, -3.883)--(0,0)); draw((2.59, -4.277)--(0,0)); dot((-3.15,-3.883)); label("A",(-3.15,-3.883),S); dot((2.59,-4.277)); label("B",(2.59,-4.277),S); dot((4.913,-0.929)); label("C",(4.913,-0.929),N); dot((0,0)); label("D",(0,0),N); markscalefactor=0.08; draw(anglemark((-3.15,-3.883),(0,0),(2.59,-4.277))); label("${2x}^\circ$",anglemark((-3.15,-3.883),(0,0),(2.59,-4.277)),S,green); draw(anglemark((-3.15,-3.883),(4.913,-0.929),(2.59,-4.277))); label("${x}^\circ$",anglemark((-3.15,-3.883),(4.913,-0.929),(2.59,-4.277)),SW,green); [/asy]


Theorem $3$: If $2$ radii form a straight line or a $180^\circ$ angle, this line is the diameter.

[asy]draw(circle((0,0),5)); draw((-5,0)--(0,0),green); draw((0,0)--(5,0),red); dot((-5,0)); label("A",(-5,0),W); dot((0,0)); label("B",(0,0),N); dot((5,0)); label("C",(5,0),E); [/asy] As seen in the picture above, radii $AB$ and $BC$ form at straight line or a $180^\circ$ angle. Therefore, segment $AC$ is a diameter of $\bigodot{B}$.

Theorem $4$:The opposite angles of a cyclic quadrilateral sum to $180^\circ$.

[asy]draw(circle((0,0),5)); [/asy]

Invalid username
Login to AoPS