There is a famous classical geometry puzzle about the angles formed by integer coordinates:
What is the sum of angle A and B in the following image? Do not use any advanced mathematics such as trigonometric formulas or complex numbers.
with an elegant answer using pure geometry (spoiler-ed for those who want to try out themselves):
Now, we consider a general problem involving angles in the form of arctan(1/n), which is the smallest angle formed by a right triangle of height 1 and width n (an integer). For example, the angle A in the above image is equal to arctan(1/3), and B is arctan(1/2).
Prove or disprove the following statement:
Given two angles α=arctan(1/n) and β=arctan(1/m) where n and m are integers, it is possible to draw three triangles ABC, DEF, and GHI on a Cartesian plane, such that all vertices have integer coordinates and the following conditions are satisfied for each triangle:
- the edge AB is parallel to the x-axis, angle A = α, angle B = β
- the edge DE is parallel to the x-axis, angle E = α, angle F = β
- the edge GH is parallel to the x-axis, angle I = α, angle G = β