# 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.

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.  $AO\cdot OB=DO\cdot OC$

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.  $PA\cdot PB=PD\cdot PC$

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.  $PA\cdot PB=PT^2$

The diameter or radius of the circle bisects the chord and its two intercepted arcs. $AB\bot CD,CE=DE,\hat{BC}=\hat{DB}$

The radius of the circle is perpendicular to the tangent line if they meet in the point of tangency.  $\angle{OTP}=90$°

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. Central Angle and Intercepted Arc

The measure of the central angle is equal to its intercepted arc.  $\theta=minor \hat{AB}$

Inscribed angle and its 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. In any cyclic quadrilateral, the product of the two diagonals is equal to the sum of the products of two opposite sides. $ac+db=d_1d_2$

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.  $\theta=\displaystyle\frac{\hat{AB}+\hat{CD}}{2}$

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. $\theta=\displaystyle\frac{|\hat{BD}-\hat{AC}|}{2}$ 