Intersecting Secants Theorem

picture info	Intersecting Secants Theorem The intersecting secant theorem or just secant theorem describes the relation of line segments created by two intersecting secants and the associated circle. For two lines ''AD'' and ''BC'' that intersect each other in ''P'' and some circle in ''A'' and ''D'' respective ''B'' and ''C'' the following equation holds: :, PA, \cdot, PD, =, PB, \cdot, PC, The theorem follows directly from the fact, that the triangles PAC and PBD are similar. They share \angle DPC and \angle ADB=\angle ACB as they are inscribed angle In geometry, an inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle. Equivalently, an in ...s over AB. The similarity yields an equation for ratios which is equivalent to the equation of the theorem given above: :\frac=\frac \Leftrightarrow , PA, \cdot, PD, =, PB, \cdot, PC, Next to the intersecting chords theo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Secant Theorem In elementary plane geometry, the power of a point is a real number that reflects the relative distance of a given point from a given circle. It was introduced by Jakob Steiner in 1826. Specifically, the power \Pi(P) of a point P with respect to a circle c with center O and radius r is defined by : \Pi(P)=, PO, ^2 - r^2. If P is ''outside'' the circle, then \Pi(P)>0, if P is ''on'' the circle, then \Pi(P)=0 and if P is ''inside'' the circle, then \Pi(P)<0. Due to the Pythagorean theorem the number $\Pi(P)$ has the simple geometric meanings shown in the diagram: For a point $P$ outside the circle $\Pi(P)$ is the squared tangential distance $, PT,$ of point $P$ to the circle $c$ . Points with equal power, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Inscribed Angle In geometry, an inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle. Equivalently, an inscribed angle is defined by two chords of the circle sharing an endpoint. The inscribed angle theorem relates the measure of an inscribed angle to that of the central angle subtending the same arc. The inscribed angle theorem appears as Proposition 20 on Book 3 of Euclid's ''Elements''. Theorem Statement The inscribed angle theorem states that an angle ''θ'' inscribed in a circle is half of the central angle 2''θ'' that subtends the same arc on the circle. Therefore, the angle does not change as its vertex is moved to different positions on the circle. Proof Inscribed angles where one chord is a diameter Let ''O'' be the center of a circle, as in the diagram at right. Choose two points on the circle, and call them ''V'' an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Intersecting Chords Theorem The intersecting chords theorem or just the chord theorem is a statement in elementary geometry that describes a relation of the four line segments created by two intersecting chords within a circle. It states that the products of the lengths of the line segments on each chord are equal. It is Proposition 35 of Book 3 of Euclid's ''Elements''. More precisely, for two chords ''AC'' and ''BD'' intersecting in a point ''S'' the following equation holds: :, AS, \cdot, SC, =, BS, \cdot, SD, The converse is true as well, that is if for two line segments ''AC'' and ''BD'' intersecting in S the equation above holds true, then their four endpoints ''A'', ''B'', ''C'' and ''D'' lie on a common circle. Or in other words if the diagonals of a quadrilateral ''ABCD'' intersect in ''S'' and fulfill the equation above then it is a cyclic quadrilateral. The value of the two products in the chord theorem depends only on the distance of the intersection point ''S'' from the circle's center a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Tangent-secant Theorem The tangent-secant theorem describes the relation of line segments created by a secant and a tangent line with the associated circle. This result is found as Proposition 36 in Book 3 of Euclid's ''Elements''. Given a secant ''g'' intersecting the circle at points G1 and G2 and a tangent ''t'' intersecting the circle at point ''T'' and given that ''g'' and ''t'' intersect at point ''P'', the following equation holds: :, PT, ^2=, PG_1, \cdot, PG_2, The tangent-secant theorem can be proven using similar triangles (see graphic). Like the intersecting chords theorem and the intersecting secants theorem The intersecting secant theorem or just secant theorem describes the relation of line segments created by two intersecting secants and the associated circle. For two lines ''AD'' and ''BC'' that intersect each other in ''P'' and some circle in '' ..., the tangent-secant theorem represents one of the three basic cases of a more general theorem about two intersecting lines and a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Power Of A Point In elementary plane geometry, the power of a point is a real number that reflects the relative distance of a given point from a given circle. It was introduced by Jakob Steiner in 1826. Specifically, the power \Pi(P) of a point P with respect to a circle c with center O and radius r is defined by : \Pi(P)=, PO, ^2 - r^2. If P is ''outside'' the circle, then \Pi(P)>0, if P is ''on'' the circle, then \Pi(P)=0 and if P is ''inside'' the circle, then \Pi(P)<0. Due to the Pythagorean theorem the number $\Pi(P)$ has the simple geometric meanings shown in the diagram: For a point $P$ outside the circle $\Pi(P)$ is the squared tangential distance $, PT,$ of point $P$ to the circle $c$ . Points with equal power, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]