Construct & Investigate
Given an arbitrary triangle ABC,
1) click on the 'Construction 1' button to construct:
.....a) the respective angle bisectors of the exterior angles at B and C
.....b) the respective perpendiculars from A to each of the angle bisectors in 1a)
.....c) label the various points as indicated in the diagram
2) click on the 'Construction 2' button to construct:
.....a) the respective angle bisectors of the interior angles at B and C
.....b) the respective perpendiculars from A to each of the angle bisectors in 2a)
.....c) label the various points as indicated in the diagram
3) What do you notice about the four constructed points D, H, I and E?
4) Check your observation in 3) carefully by dragging. Does the result hold if ABC becomes obtuse?
Conject
5) Formulate a conjecture of your observations in 3) and 4).
Four Collinear Points
Conjecture
You should've noticed that the four constructed points D, H, I and E are collinear (lie in a straight line) on a line parallel to BC.
Challenge
6) Can you explain why (prove that) your conjecture is true? Can you prove it in more than one way?
Hints: i) It often helps to construct auxiliary lines and points - click on the 'Show Auxiliary Objects' button.
...........ii) Next have a look at Triangle Midpoints and/or Interesting Locus Result.
Explore More
7) Click on the 'Show CircumCircles' button to construct the respective circumcircles of triangle ADB and AEC.
8) What do you notice about the circumcircles in 7)? Check your observation by dragging.
Challenge 1
9) Can you explain/prove your observations about the circumcircles?
Hints: i) What do you notice about quadrilaterals ADBI and AHCE? What type of quadrilaterals are they? Why?
...........ii) Alternatively, click on the 'Show Hint' button and use the properties of cyclic quadrilaterals.
10) Click on the 'Show Tangent Circles' button to construct two circles with respective diameters HI and DE.
11) What do you notice about the relationship between the radii of the four circles? Check by dragging.
Challenge 2
12) Can you explain/prove your observations about the radii of the four circles?
Submitted Paper
A paper by Moshe Stupel (Israel) and myself entitled "Four Collinear Points" has been submitted to the Learning & Teaching Mathematics journal of AMESA. All rights reserved.
Related Links
Reasoning Backwards: Triangle Midpoints (Rethinking Proof activity)
Cyclic Quadrilateral Converse (Rethinking Proof activity)
Interesting Locus Result
Collinear Conjecture
Isosceles Triangle Collinear Conjecture
Triangle Incentre-Circumcentre Collinearity
Some Trapezoid (Trapezium) Explorations (see Investigations 3 & 4)
Parallelogram Distances (prove F, P, G and H, P, I, respectively, are collinear)
An interesting collinearity
Euler line proof
Further Euler line generalization
Euler-Nagel line analogy
Nine Point Conic and Generalization of Euler Line
Spieker Conic and generalization of Nagel line
More Properties of a Bisect-diagonal Quadrilateral
Quadrilateral Similar Triangles Collinearity
Investigating incentres of some iterated triangles
Investigating circumcentres of iterated median triangles
Concurrency, collinearity and other properties of a particular hexagon
External Links
Collinearity (Wikipedia)
Midpoint theorem (triangle) (Wikipedia)
Cyclic quadrilateral (Wikipedia)
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.)
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Michael de Villiers, created with WebSketchpad, 27 Jan 2026; updated 30 Jan 2026.