Investigate
Given a quadrilateral ABCD, with E, F, G and H the respective midpoints of AB, BC, CD and DA. Label as I and J the respective intersections of EG with diagonals BD and AC, and K and L the respective intersections of HF with diagonals BD and AC.
1) Drag any of the red vertices A, B, C or D and carefully look at the displayed values.
2) What do you notice? Can you formulate a conjecture?
Challenge
3) Can you explain why (prove that) your conjecture(s) above are true? Only if stuck, click on the provided Show Hint.
4) From the above result, can you give two more pairs of ratios that are equal? Check your answer by clicking on Show Other Ratios, and by further dragging.
Note: Observe by dragging that the result is also true if ABCD is concave as well as crossed as shown below.
Some Related Links
Kite Midpoints (Generalizes to Varignon's theorem - Rethinking Proof activity)
Point Mass Centroid (centre of gravity or balancing point) of Quadrilateral
A diagonal property of a Rhombus constructed from a Rectangle
Area Formula for Quadrilateral in terms of its Diagonals
Relations between the sides and diagonals of parallelo-hexagons
The quasi-circumcentre and quasi-incentre of a quadrilateral
Unproved Conjecture - Quasi-Euler Inequality for Quadrilateral
More Properties of a Bisect-diagonal Quadrilateral
Experimentally Finding the Centroid of a Triangle with Different Weights at the Vertices (Ceva's Theorem)
The Center of Gravity of a Triangle (Rethinking Proof activity, incl. Ceva's theorem)
External Links
Menelaus's theorem (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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Free Download of Geometer's Sketchpad & Associated Learning/Instructional Modules on Various Topics
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