Find the four angles of a cyclic quadrilateral abcd in which

The angles of cyclic quadrilaterals satisfy several important relations, as they are all \[\begin\), or \(\pi\) radians. If \(ABCD\) is a cyclic quadrilateral, find the value of \(\cos = \angle AOB\), where \(O\) is the center of the circle, by the inscribed angle theorem. This can also lead to useful information, if the center of the circumcircle is relevant. Consider all sets of 4 points \(A, B, C, D \) which satisfy the following conditions: • \(AB\) is an integer. • \(BC = AB + 1 \). • \(CD = BC + 1\). • \( DA = CD + 1\). • \(AC= DA + 1\) • \( AC \) divides \( AB \times CD + BC \times DA \). Over all such sets, what is \( \max \lceil BD \rceil ? \) In fact, it is true of any quadrilateral that \[AB \cdot CD \leq AC \cdot BD + BC \cdot AD,\] meaning that the cyclic quadrilateral is the equality case of this inequality. In fact, more can be said about the diagonals: if \(a,b,c,d\) are the lengths of the sides of the quadrilateral (in clockwise order), \[\begin.\] Here are few well-known problems which use the basic properties of cyclic quadrilaterals. They are mainly of Olympiad flavor and are solvable by elementary methods. Problem 1. Let \(E\) and \(F\) be two points on side \(BC\) and \(CD\) of square \(ABCD\), such that \(\angle EAF=\ang\). Let \(M\) and \(N\) be the intersection of diagonal \(BD\) with \(AE\) and \(AF,\) respectively. Let \(P\) be the intersection of \(MF\) and \(NE\). Prove that \(AP\) is perpendicular to \(EF\). Problem 2. \(\triangle ABC\) is in...

Cyclic Quadrilateral A cyclic quadrilateral is a four-sided polygon inscribed in a circle. It has the maximum area possible with the given side lengths. In other words, a quadrilateral inscribed in a circle depicts the maximum area possible with those side lengths. Let us learn more about a cyclic quadrilateral and its properties in this article. 1. 2. 3. 4. 5. Cyclic Quadrilateral Definition A cyclic quadrilateral means a quadrilateral that is inscribed in a The word "cyclic" is from the Greek word "kuklos", which means "circle" or "wheel". The word "quadrilateral" is derived from the ancient Latin word "Quadri", which means "four-side" or "latus". In the figure given below, ABCD is a cyclic quadrilateral with a, b, c, and d as the side-lengths and p and q as the diagonals. Properties of Cyclic Quadrilateral The properties of a cyclic quadrilateral help us to identify this figure easily and to solve questions based on it. Some of the properties of a cyclic quadrilateral are given below: • In a cyclic quadrilateral, all the four vertices of the quadrilateral lie on the • The four sides of the inscribed quadrilateral are the four • The measure of an exterior angle at a vertex is equal to the opposite interior angle. • In a cyclic quadrilateral, p × q = sum of product of opposite sides, where p and q are the diagonals. • The • The perpendicular bisectors of the four sides of the cyclic quadrilateral meet at the center O. • The sum of a pair of opposite angles is 180° (supple...

\( \newcommand\), then \(ABCD\) is inscribed in \(\bigodot E\). What if you were given a circle with a quadrilateral inscribed in it? How could you use information about the arcs formed by the quadrilateral and/or the quadrilateral's angle measures to find the measure of the unknown quadrilateral angles? Example \(\PageIndex\) Review Fill in the blanks. • A(n) _______________ polygon has all its vertices on a circle. • The _____________ angles of an inscribed quadrilateral are ________________. Quadrilateral \(ABCD\) is inscribed in \(\bigodot E\). Find: Figure \(\PageIndex\) Vocabulary Term Definition central angle An angle formed by two radii and whose vertex is at the center of the circle. chord A line segment whose endpoints are on a circle. circle The set of all points that are the same distance away from a specific point, called the center. diameter A chord that passes through the center of the circle. The length of a diameter is two times the length of a radius. inscribed angle An angle with its vertex on the circle and whose sides are chords. intercepted arc The arc that is inside an inscribed angle and whose endpoints are on the angle. radius The distance from the center to the outer rim of a circle. Inscribed Polygon An inscribed polygon is a polygon with every vertex on a given circle. Inscribed Quadrilateral Theorem The Inscribed Quadrilateral Theorem states that a quadrilateral can be inscribed in a circle if and only if the opposite angles of the quadrilater...

ABCD is a cyclic quadrilateral ∠A + ∠C = 180° ......(Sum of the opposite angles of a cyclic quadrilateral is 180°) (4y + 20)° + (4x)° = 180° 4y + 20 + 4x = 180 4x + 4y = 180 – 20 4x + 4y = 160 x + y = 40 → (1) ...(divided by 4) ∠B + ∠D = 180° ...(Sum of the opposite angles of a cyclic quadrilateral) (3y – 5)° + (7x + 5)° = 180° 3y – 5 + 7x + 5 = 180 7x + 3y = 180 → (2) (1) × 3 ⇒ 3x + 3y = 120 → (3) (3) – (2) ⇒– 4x = – 60 4x = 60 x = `60/4` Substitute the value of x = 15 in (1) 15 + y = 40 y = 40 – 15 = 25 ∠A = 4y + 20 = 4(25) + 20 = 100 + 20 = 120° ∴∠A = 120° ∠B = 3y – 5 = 3(25) – 5 = 75 – 5 = 70 ∴∠B = 70° ∠C = 4x = 4(15) = 60 ∴∠C = 60 ∠D = 7x + 5 = 7(15) + 5 ∠D = 105 + 5 = 110° ∴∠A = 120°, ∠B = 70°, ∠C = 60° and ∠D = 110°