Given and two squares in Fig. 1. The squares are sitting on the two sides of and , respectively. Both squares are oriented away from the interior of . is an isosceles right triangle. is on the same side of . Prove that points and lie on the same line.
Introducing rectangular coordinates show in Fig. 2:
We observe that
Solving system of equations (4), (5), (6), (7), we obtain four set of solutions:
Among them, only (10) truly represents the coordinates in Fig. 2.
By Heron’s formula (see “Had Heron Known Analytic Geometry…“), the area of triangle with vortex is
From Fig. 4, we see that it is zero.
lie on the same line.
The reason we do not consider (9), (10), (11) is due to the fact that
(9) contradicts (2) since .
by (1), (10) and (11) indicate which contradicts (3).
Exercise-1 Prove “ lie on the same line” with complex numbers (hint: see “Treasure Hunt with Complex Numbers“).