The project aims at finding the location of the fix point of two similar figures by using the concept of Brouwer’s Fixed Point Theorem. Based on knowledge of similar triangles and concyclic quadrilaterals, the team proved that the fix point is the point of intersection of three circumcircles constructed on the sides of the triangles, where each of the circles passes through one pair of corresponding vertices, and the point of intersection between two extended sides of the triangle. The method was extended to parallelograms and trapeziums, and other special cases.

This project focuses on the investigation of Pick’s formula, how to proof the original Pick’s formula on square lattice system. The project also conduct proof on the extension of Pick’s formula on isometric lattice system. This project has also done a generalization of Pick’s formula for polygons with non-touching holes. The common use of isometric system on the projection of solids draws the attention on the study of the application of Pick’s formula on crystallography.

The project aims at constructing a foundation of junior secondary pure geometry suitable to read for junior students or elementary geometry learners. Being different from but also inspired by the systems in Euclid’s Elements and Hilbert’s Foundations of Geometry, the team have strived to keep our work easy to understand intuitively but still sufficiently rigorous to prove the results regarding straight lines, parallel lines and congruent triangles, rather than aiming to lay a foundation with the least number of axioms possible as in these two classics.

Please refer to the Chinese webpage

Please refer to the Chinese webpage

Please refer to the Chinese webpage

We designed a formula to give a percentage that reflects the chance of admission to the 20 university programs. Afterwards, we evaluated the overall competitiveness of the university application by taking into account the advice given to applicants as to what makes a good application. We shall then give suggestions for improving the application. Finally, we made a comparison on whether it is easier to enter a university course via JUPAS (with HKDSE results) or non-JUPAS (with GCE A Level results).