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Part of 1964 Vietnam National Olympiad

Problems(1)

old vietanmese locus in space

Source: Vietnamese MO (VMO) 1964

8/22/2018
Let PP be a plane and two points A∈(P),O∉(P)A \in (P),O \notin (P). For each line in (P)(P) through AA, let HH be the foot of the perpendicular from OO to the line. Find the locus (c)(c) of HH. Denote by (C)(C) the oblique cone with peak OO and base (c)(c). Prove that all planes, either parallel to (P)(P) or perpendicular to OAOA, intersect (C)(C) by circles. Consider the two symmetric faces of (C)(C) that intersect (C)(C) by the angles α\alpha and β\beta respectively. Find a relation between α\alpha and β\beta.
geometry3-Dimensional GeometryLocus