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1/ OM + 1/ ON + 1/OP is fixed, trihedral angle related

Source: Polish MO second round 1955 p6

August 29, 2024
3D geometrygeometryfixed

Problem Statement

Inside the trihedral angle OABC OABC , whose plane angles AOB AOB , BOC BOC , COA COA are equal, a point S S is chosen equidistant from the faces of this angle. Through point S S a plane is drawn that intersects the edges OA OA , OB OB , OC OC at points M M , N N , P P , respectively. Prove that the sum 1OM+1ON+1OP \frac{1}{OM} + \frac{1}{ON} + \frac{1}{OP} has a constant value, i.e. independent of the position of the plane MNP MNP .
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