Side Divider Theorem for a Circumscribed/Tangential Quadrilateral
For a proof of the quadrilateral case, read my AMESA 1996 paper An interesting Side-Angle Duality in Geometry.
For a side-angle dual of this result involving the angle dividers of a cyclic quadrilateral go to Angle Divider Theorem for a Cyclic Quadrilateral.
For dynamic versions of the Angle & Side Divider cases for a Triangle go to Conway's Circle Theorem as special case of Side Divider Theorem.
Proofs of this result, its dual & of the generalizations, are given in De Villiers (1994, 1996, 2009) as well as generalizations to circumscribed polygons with an even or odd number of sides.
Reference
De Villiers, M. (1994, 1996, 2009). Some Adventures in Euclidean Geometry. (Free to download). Morrisville, NC: Lulu Publishers, Dynamic Mathematics Learning.
Related Links
Cyclic Hexagon Alternate Angles Sum Theorem
Circumscribed Hexagon Alternate Sides Theorem
A generalization of the Cyclic Quadrilateral Angle Sum theorem
Angle Divider Theorem for a Cyclic Quadrilateral
More Area, Perimeter and Other Properties of Circumscribed Isosceles Trapeziums and Cyclic Kites (PDF)
Some Trapezoid (Trapezium) Explorations
Midpoint trapezium (trapezoid) theorem generalized
Tiling with a Trilateral Trapezium and Penrose Tiles (PDF)
Matric Exam Geometry Problem - 1949
Finding the Area of a Crossed Quadrilateral
The Parallel-pentagon and the Golden Ratio
International Mathematical Talent Search (IMTS) Problem Generalized
Clough's Theorem (a variation of Viviani) and some Generalizations
A Geometric Paradox Explained
Semi-regular Angle-gons and Side-gons: Generalizations of rectangles and rhombi
Intersecting Circles Investigation
SA Mathematics Olympiad Problem 2016, Round 1, Question 20
SA Mathematics Olympiad 2022, Round 2, Q25
An extension of the IMO 2014 Problem 4
A 1999 British Mathematics Olympiad Problem and its dual
Dirk Laurie Tribute Problem
Golden Quadrilaterals
Crossed Quadrilateral Properties
Extangential Quadrilateral
Triangulated Tangential Hexagon theorem
Theorem of Gusić & Mladinić
Conway's Circle Theorem as special case of Side Divider (Windscreen Wiper) Theorem
Pirate Treasure Hunt and a Generalization
A Quarter Circle Investigation, Explanation & Generalization
External Links
Quadrilateral (Wikipedia)
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
UCT Mathematics Competition Training Material
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)
***************
Free Download of Geometer's Sketchpad & Learning/Instructional Modules on various topics
***************
Back to "Dynamic Geometry Sketches"
Back to "Student Explorations"
Created by Michael de Villiers in 2011 using JavaSketchpad, updated 24 March 2012; updated 3 September 2020 to WebSketchpad; updated 2 March 2024; 12 May 2026.