Theorems on Circle
I compiled some useful theorems on circle and here’s what I got. Trust me; these theorems are very helpful when you are working with circles. Memorizing or familiarization of these formulas will make your life easier when you’re dealing with circles.
You can apply these concepts in this page.
Intersecting chords inside the Circle
The product of the lengths of the segment of the chords of one chord is equal to the product of the length of the segment of the other chord.
Intersecting secants inside the Circle
The product of the length of one the entire secant and its external length is equal to the product of the lengths of the other secant and its external length.
Tangent and Secant line intersecting outside the circle
The product of the lengths of the entire secant line and its external length is equal to the square of the length of the tangent line.
Diameter/Radius as perpendicular bisector
The diameter or radius of the circle bisects the chord and its two intercepted arcs.
Radius and tangent line perpendicularity
The radius of the circle is perpendicular to the tangent line if they meet in the point of tangency.
Angles Subtended to the Same Arc
Angles are congruent if they intercepted to the same arc.
Given arc CD is the common arc,
If two circles are tangent with each other, the line connecting their centers passes through the point tangency.
Central Angle and Intercepted Arc
The measure of the central angle is equal to its intercepted arc.
Inscribed angle and its intercepted arc
Angle Inscribed in a semicircle
Any angle inscribed in a circle with its two rays connecting the endpoints of the diameter is a right angle. We can also say that a triangle with one side as the circle’s diameter is a right triangle.
Wherever is the position of point D. It is will always form a right angle.
Cyclic Quadrilaterals and Ptolemy’s Theorem
In any cyclic quadrilateral, the product of the two diagonals is equal to the sum of the products of two opposite sides.
Two tangent lines on the same circle that intersect the same external point is the equal.
In the figure,
Angle formed by Intersecting Chords
The angle formed by the intersection of two chords inside the circle is half the sum of the intercepted arcs.
Angle formed by intersecting secants
If the two secants meet outside the circle, the angle made is half the absolute value of the difference of two intercepted arcs.