MathDB

4.1

Part of 2011 Saudi Arabia Pre-TST

Problems(2)

distance between 2 neighboring stations on Earth

Source: 2011 Saudi Arabia Pre-TST November 4.1 https://artofproblemsolving.com/community/c2745403_2011_

1/1/2022
A Geostationary Earth Orbit is situated directly above the equator and has a period equal to the Earth’s rotational pe­riod. It is at the precise distance of 22,23622,236 miles above the Earth that a satellite can maintain an orbit with a period of rotation around the Earth exactly equal to 2424 hours. Be­ cause the satellites revolve at the same rotational speed of the Earth, they appear stationary from the Earth surface. That is why most station antennas (satellite dishes) do not need to move once they have been properly aimed at a tar­ get satellite in the sky. In an international project, a total of ten stations were equally spaced on this orbit (at the precise distance of 22,23622,236 miles above the equator). Given that the radius of the Earth is 39603960 miles, find the exact straight dis­tance between two neighboring stations. Write your answer in the form a+bca + b\sqrt{c}, where a,b,ca, b, c are integers and c>0c > 0 is square-free.
geometry
|PM-PN|/PC is constant , MC _|_ NC, semicircle of diameter AB and center C

Source: 2011 Saudi Arabia Pre-TST February 4.1 https://artofproblemsolving.com/community/c2745403_2011_

1/1/2022
On a semicircle of diameter ABAB and center CC, consider vari­able points MM and NN such that MC⊥NCMC \perp NC. The circumcircle of triangle MNCMNC intersects ABAB for the second time at PP. Prove that ∣PM−PN∣PC\frac{|PM-PN|}{PC} constant and find its value.
ratiofixedgeometrysemicircle