Question

prove that rectangles dont exist in hyperbolic geometry

Answer 1

In hyperbolic geometry, rectangles do not exist and all triangles have angle less than 180. All convex quadilaterals have angle sum less than 360.

Proof-

1) Draw a line l, drop a line perpendicular to l PQ.

2) Draw a line m through P perpendicular to PQ.

3) Let R be any point on l, and draw a perpendicular t to l through R.

4) Let S be the foot of perpendicular to t through p.

5) The line PS does not intersect l since both are perpendicular to t and PS is not equal to m.

6) If S is straight line , then PQRS is a rectangle.

7) In hyperbolic geometry if one rectangle exists all triangles have defect 0.

8) Point (7) is a cotradiction to statement that "In hyperbolic geometry all triangle have sum of angle less than 180.

9) Hence, PS will never be equal to m, which proves that rectangles do not exist in hyperbolic geometry.