What Is Angle Chasing?
Angle chasing is a popular method of solving problems in geometry that employs various properties of, and identities that include angles.
Angles are basic elements in many geometric situations; there are inscribed, secant, complementary, vertical, supplementary angles.
Equal angles emerge as
inscribed into the same circle and subtended by the same or equal arcs,
two halves of an angle divided by the angle bisector,
corresponding angles in equal or similar triangles or other shapes,
corresponding or alternate angles in parallel lines.
It's worth remembering that
Acute angles in a right triangle are complementary; they add up to 90°.
Angles formed by a straight line and a ray emanating from one of its points are supplementary; they add up to 180°.
The sum of angles in a (plane) triangle is always 180°.
Central angles in a circle are twice as large as the inscribed angles subtend by the same arc.
The angles at the base of an isosceles triangle are equal. (Thales' Theorem>)
A tangent to a circle is perpendicular to the radius at the point of tangency.
The Exterior Angle Theorem: an angle exterior to a triangle equals the sum of two enterior and opposite angles.
Inscribed and secant angles are nicely expressed interms of subtending arcs.
And, of course, All right angles are equal.
Here is an incomplete list of problems at this site where angle chasing plays a major role in the solution:
- A Family of Cyclic Quadrilaterals
- A Property of Circle Through the Incenter
- Angle Chasing on Radical Axis
- Angle in Square
- Angles Inscribed in an Absent Circle
- Chain of Four Intersecting Circles
- Chasing Angles Among Angle Bisectors
- Chasing Angles in Pascal's Hexagon
- Chasing Secant Angles
- Circle Chains and Inscribed Angles
- Circle through the Incenter
- Circumcenter and Orthocenter
- Concyclic Points from Midpoint of an Arc
- Concyclicity from Collinearity
- Cyclic Quadrilateral, Angle Bisector And Isosceles Triangle
- From Perpendicular Center Lines to Concyclic Points
- Garcia's Two Circles Lemmas
- Garcia-Feuerbach Collinearity
- Golden Ratio in a Mutually Beneficial Relationship
- Integer Triangles with Two Angles in the Ratio 1 : 2
- Isogonal Lemma
- K. Knop's Problem with Two Regular Pentagons And an Equilateral Triangle
- Miquel Circumcenters
- On Bottema's Shoulders with a Ladder
- Problem 1 from the EGMO2017
- Problems of Apollonius: An Elementary Solution in the PPL case
- Pure Angle Chasing
- Pure Angle Chasing II
- Pure Angle Chasing III
- Reflections of a Line Through the Orthocenter
- Similar Triangles in Crossing Circles
- Simple Property of Circle Through the Incenter
- Stereographic Projection of a Coffin Problem
- The 80-80-20 Triangle
- Two Circles In a Square II
- Two Circles, Two Segments - One Ratio
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