Circle Theorems

Intro to Circle Theorems

Part #1:

For any angle in a triangle above the radius, the angle that touches the surface of the circle has to equal 90°.[br]In this example, the angle [math]\alpha[/math] will equal 90° as long as it is above the CB radius.[br]Move the point D along circle surface to check.

Part #2:

Inscribed Angle Theorems:[br]An inscribed angle a° is half of the central angle 2a°[br]Here we have the angle [math]\alpha[/math] is half of the central angle [math]\beta[/math]. Even if you moved the point [img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA4AAAAMCAYAAABSgIzaAAAAAXNSR0IArs4c6QAAAMVJREFUKFOd0r9KgmEUx/HPu3UJOTdG0RWIOeXUEF5AQ4tCg4EEDbo6eAXi1hQECja0NkhLRkO0NHQTzsoDj/Dw8r4S75kOh/M9v/MvU9GyipwycIAuvvGJXhSo4y34ebCGOUZ4jskTvGOKMe6KwCGaCJV3dotjPGKNVRH4hQUeErCFPn7Q2cXzrW5wgdcEbCDMfI2/MjAs4xK/CXgTF3WWXiCv+IQ2znGKq6j2gnss8VE0Y4id4AizROEQB/ta/fc/VP6cLarYHw08EHmnAAAAAElFTkSuQmCC[/img] along the circle between the points B and D, you would still get that the angle [img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA0AAAATCAYAAABLN4eXAAAAAXNSR0IArs4c6QAAAPlJREFUOE+d07ErhlEUx/HPazMqo8ViMFjIYmFTZizKxEZSDCZ/gUWKDEoZsMpisCll4A9QSiblLxA6el7dbvd5nry37nLv+f7O755zbkcPq9MDow06x0Ii/BvfBH1jH+sVdIp7HNRB87jIRK9xFrsO+sIiLqssY3jqipSgcTxUAceYREDLCIvFNx2hDysIm5FhC+/YqYOiAEN4S6o2jRMMN0G57Q0sYaIEDeCjYPuqeuduCdpG7MHE2ihuMIWXEvSIfoxgBp84xBpuu0K59yhCnIV6NPcOq/l85tAeNtuGOIVmMVdZaeRSKHrxjNf/ZGqL/btv+09FoR8PXygUPvY/9QAAAABJRU5ErkJggg==[/img] is twice [img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA4AAAAMCAYAAABSgIzaAAAAAXNSR0IArs4c6QAAAMVJREFUKFOd0r9KgmEUx/HPu3UJOTdG0RWIOeXUEF5AQ4tCg4EEDbo6eAXi1hQECja0NkhLRkO0NHQTzsoDj/Dw8r4S75kOh/M9v/MvU9GyipwycIAuvvGJXhSo4y34ebCGOUZ4jskTvGOKMe6KwCGaCJV3dotjPGKNVRH4hQUeErCFPn7Q2cXzrW5wgdcEbCDMfI2/MjAs4xK/CXgTF3WWXiCv+IQ2znGKq6j2gnss8VE0Y4id4AizROEQB/ta/fc/VP6cLarYHw08EHmnAAAAAElFTkSuQmCC[/img][br]Move the point C along circle surface to check.

Part #3:

Angles Subtended by Same Arc Theorem:[br]Notice that the angle C equals the angle E. This Theorem states that these two angles will always be equal to each other as long as they are in the range from B to D[br]Move the point C and E along circle surface to check.

Part #4:

Cyclic Quadrilateral[br]A Cyclic Quadrilateral has every vertex on a circle's circumference:[br]A Cyclic Quadrilateral opposite angles add to 180°.[br]In this example we have [math]\alpha[/math]+[math]\beta[/math]=180°[br]And [math]\gamma[/math]+[math]\delta[/math]=180°[br]Move the points along circle surface to check.