चक्रिय चतुर्भुज
A cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. Other names for the cyclic quadrilaterals are inscribed quadrilateral , concyclic quadrilateral and chordal quadrilateral, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals.
The formulas and properties discussed here are only valid in the convex case.
The word cyclic is from the Ancient Greek κύκλος (kuklos), which means "circle" or "wheel".
Please note that, all triangles have a circumcircle, but not all quadrilaterals do. An example of a quadrilateral that cannot be cyclic is a non-square rhombus.
चक्रीय चतुर्भुज एक चतुर्भुज हो जसको सबै शीर्षबिन्दुहरु एउटै वृत्तमा हुन्छन् । सर्कललाई सर्कमसर्कल वा सर्कमस्क्राइब्डसर्कल भनिन्छ र शीर्षबिन्दुहरुलाई चक्रीय शीर्षबिन्दु भनिन्छ। चक्रीय चतुर्भुज लाई इस्न्क्राइब्ड, कन्साइक्लिक र कोर्डल चतुर्भुज पनि भनिन्छ। सामान्यतया चक्रीय चतुर्भुज कन्भेक्स हुन्छ, तर यो क्रस गरिएको पनि हुन्छन।
तर यस पाठमा चर्चा गरिएका कुराहरु कन्भेक्स चक्रीय चतुर्भुजमा मात्र मान्य हुनेछ।
चक्रीय शब्द प्राचीन ग्रीक भाषाको κύκλος (kuklos) बाट आएको हो, जसको अर्थ "वृत्त" वा "चक्र" भन्ने हुन्छ।
नोटः सबै त्रिभुजको परिधि वृत हुन्छ, तर सबै चतुर्भुजको परिधि वृत हुँदैनन् जस्तै वर्ग नभएको समलम्ब चतुर्भुज ।
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 circumference of the circle.
- The four sides of the inscribed quadrilateral are the four chords of the circle.
- In a cyclic quadrilateral, p × q = sum of product of opposite sides, where p and q are the diagonals.
- The perpendicular bisectors are always concurrent.
- 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° (supplementary). Let ∠A, ∠B, ∠C, and ∠D be the four angles of an inscribed quadrilateral. Then, ∠A+∠C=180° and ∠B+∠D=180°.
- The measure of an exterior angle at a vertex is equal to the opposite interior angle.
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