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.

Chord-chord theorem

AO\cdot OB=DO\cdot OC

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.

secant-secant theorem

PA\cdot PB=PD\cdot PC

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.

secant-tangent theorem

PA\cdot PB=PT^2

Diameter/Radius as perpendicular bisector

The diameter or radius of the circle bisects the chord and its two intercepted arcs.

diameter as perpendicular bisector

AB\bot CD,CE=DE,\hat{BC}=\hat{DB}

Radius and tangent line perpendicularity

The radius of the circle is perpendicular to the tangent line if they meet in the point of tangency.

radius-tangent perpendiculity


Angles Subtended to the Same Arc

Angles are congruent if they intercepted to the same arc.


Given arc CD is the common arc, \angle{CAD}=\angle{CBD}

Tangent Circles

If two circles are tangent with each other, the line connecting their centers passes through the point tangency.

Internal and external tangents

Central Angle and Intercepted Arc

The measure of the central angle is equal to its intercepted arc.

Central angle and intercepted arc

\theta=minor \hat{AB}

Inscribed angle and its intercepted arc

inscribed angle and intercepted arc

m\angle{\hat{AB}}=2\cdot m\angle{ADB}

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.

angle in a semicircle

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.

ptolemy's theorem


Intersecting Tangents

Two tangent lines on the same circle that intersect the same external point is the equal.

In the figure, PA=PB


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 by intersecting chord


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.


intersecting secants

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