Draw a circle theory

• Drawing: Draw a central point (the center). Use a compass to draw a perfectly round curve around this center.

• Drawing: Draw a line segment from the center to any point on the circle.

• Drawing: Draw a line segment through the center, connecting two points on the circle.

• Drawing: Draw a line segment connecting two points on the circle without necessarily passing through the center.

• Drawing: Highlight a section of the circle's curve between two points.

• Drawing: Draw a line that just grazes the circle at a single point.

Angle at the Center Theorem: The angle subtended by an arc at the center of a circle is twice the angle subtended by the same arc at any point on the remaining part of the circumference.

• Drawing: Draw a circle with center O. Mark two points A and B on the circumference. Draw lines OA and OB to form $\angle AOB$ (angle at the center). Mark another point C on the circumference (on the major arc AB). Draw lines AC and BC to form $\angle ACB$ (angle at the circumference). You will observe that $\angle AOB = 2 \times \angle ACB$.

Angles in the Same Segment Theorem: Angles subtended by the same arc in the same segment of a circle are equal.

• Drawing: Draw a circle. Mark two points A and B on the circumference to define an arc AB. Mark two other points C and D on the circumference within the same segment (e.g., on the major arc). Draw lines AC, BC, AD, and BD. You will find that $\angle ACB = \angle ADB$.

Angle in a Semicircle Theorem: The angle subtended by a diameter at any point on the circumference is a right angle ($90^\circ$).

• Drawing: Draw a circle with center O. Draw a diameter AB. Mark any point C on the circumference. Draw lines AC and BC. The angle $\angle ACB$ will always be $90^\circ$.

Tangent-Radius Property: The tangent at any point of a circle is perpendicular to the radius through the point of contact.

• Drawing: Draw a circle with center O. Draw a tangent line to the circle at a point P on the circumference. Draw the radius OP. The angle between the tangent line and the radius OP at point P will be $90^\circ$.

Cyclic Quadrilateral Theorem: The opposite angles of a cyclic quadrilateral (a quadrilateral whose vertices all lie on the circumference of a circle) sum to $180^\circ$.

• Drawing: Draw a circle. Mark four points A, B, C, and D on the circumference. Connect these points in order to form a quadrilateral ABCD. You will find that $\angle A + \angle C = 180^\circ$ and $\angle B + \angle D = 180^\circ$.