Let l be a Poincare line and let P be a point not on l. Drop a perpendicular from P to l. Let Q be the foot of this perpendicular. Let A and B be two points on l on opposite sides of Q. Let the hyperbolic distance from P to Q be d. Taking the hyperbolic constant, k, to be equal to 1, prove that the measure of angle APB must be less than 4arctan(e^-d).

Let l be a Poincare line and let P be a point not on l. Drop a perpendicular from P to l. Let Q be the foot of this perpendicular. Let A and B be two points on l on opposite sides of Q. Let the hyperbolic distance from P to Q be d. Taking the hyperbolic constant, k, to be equal to 1, prove that the measure of angle APB must be less than 4arctan(e^-d).