Circle theorems | How to Solve Challenging MATHs problems and questions | Geometry Hard level

Circle theorems | How to Solve Challenging MATHs problems and questions | Geometry Hard level

This is very interesting problem dedicated to Circle Theorems : Questions and Problems.

This topic is a subtopic for Circle Geometry, so hope you are familiar with Circle properties and inscribed, exterior, central angles properties.

GCSE MATHs Circle theorems questions | GCSE Higher MATHs 2020 Circle Problem 1 Medium level


Quick Reminder:

CIRCLE THEOREMS

A circle is the set of all points (LOCI) which are equidistant from a fixed point called the centre

The circumference is the distance or the length around the entire circle boundary.

An arc of a circle is any continuous part of the circle. A chord of a circle is a line segment joining any two points on the circle. A radius of a circle is any line segment joining its centre to any point on the circle. A diameter of a circle is any chord passing through its centre.

A semicircle is a half of a circle.

A tangent to a circle is any line which touches the circle in exactly one point.

CIRCLE THEOREMS

Tangents from an external point: Tangents from an external point are equal in length.
Radius and a tangent: The tangent to a circle is perpendicular to the radius at the point of contact (common point)

Chords of a circle: The perpendicular from the centre of a circle to a chord bisects the chord.

Angle in a semi-circle: The angle in a semicircle is a right angle.

If a circle can be drawn through four points we say that the points are concyclic and by joining them we can form a convex quadrilateral. That quadrilateral becomes a cyclic quadrilateral.

CYCLIC QUADRILATERAL THEOREMS

Opposite angles of a cyclic quadrilateral: The opposite angles of a cyclic quadrilateral supplementary, or add to 180.
Exterior angle of a cyclic quadrilateral: The exterior angle of a cyclic quadrilateral is equal to the opposite interior angle.

Optimization
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