Locus of right angles

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Source: IMO 1963, Day 1, Problem 2

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1398 posts

DPopov

#1 h Oct 14, 2005, 1:16 AM • 1 Y

Y by Adventure10

Point $A$ and segment $BC$ are given. Determine the locus of points in space which are vertices of right angles with one side passing through $A$ , and the other side intersecting segment $BC$ .

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2643 posts

yetti

#2 h Oct 30, 2005, 2:54 AM • 2 Y

Y by Adventure10, Mango247

Let F be the foot of a normal from the point A to the line BC and P an arbitrary point on the segment BC. If the point A does not lie on the line BC, the locus of the vertices V of the right angles $\angle AVP$ is a sphere with a diameter AP and passing through the point F. This sphere is centered on the midline M, N of the triangle $\triangle ABC$ parallel to the side BC, where M, N are the midpoints of the sides AB, AC. If the point A lies on the line BC, then the points A, F and the locus of the vertices V of the right angles $\angle AVP$ is a sphere centered on segment M, N, where M, N are the midpoints of the segments AB, AC. As the point P moves on the segment BC, the locus of the vertices V becomes a portion of the pencil of spheres centered on the segment MN and passing through the points A, F. If the point A does not lie on the line BC, the pencil of spheres is elliptic (the spheres intersect in a fixed circle with the diameter AF and perpendicular to the line BC). If the point A lies on the line BC, the pencil of spheres is parabolic (the spheres touch the plane perpendicular to the line BC at the point A).

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#3 h Jul 15, 2022, 2:25 PM • 1 Y

Y by hungrypig

Let the intersection of the right angle with $BC$ be $P.$ Therefore, given $A$ and $P,$ all possible vertices of the right angle lie on the circle with diameter $AP.$ Thus, when all possible circles are drawn, it encloses a region as shown in the diagram below:

$[asy] draw(circle((3.500000,3.000000),4.609772)); draw(circle((3.550000,3.000000),4.571925)); draw(circle((3.600000,3.000000),4.534314)); draw(circle((3.650000,3.000000),4.496943)); draw(circle((3.700000,3.000000),4.459821)); draw(circle((3.750000,3.000000),4.422952)); draw(circle((3.800000,3.000000),4.386342)); draw(circle((3.850000,3.000000),4.350000)); draw(circle((3.900000,3.000000),4.313931)); draw(circle((3.950000,3.000000),4.278142)); draw(circle((4.000000,3.000000),4.242641)); draw(circle((4.050000,3.000000),4.207434)); draw(circle((4.100000,3.000000),4.172529)); draw(circle((4.150000,3.000000),4.137934)); draw(circle((4.200000,3.000000),4.103657)); draw(circle((4.250000,3.000000),4.069705)); draw(circle((4.300000,3.000000),4.036087)); draw(circle((4.350000,3.000000),4.002812)); draw(circle((4.400000,3.000000),3.969887)); draw(circle((4.450000,3.000000),3.937321)); draw(circle((4.500000,3.000000),3.905125)); draw(circle((4.550000,3.000000),3.873306)); draw(circle((4.600000,3.000000),3.841875)); draw(circle((4.650000,3.000000),3.810840)); draw(circle((4.700000,3.000000),3.780212)); draw(circle((4.750000,3.000000),3.750000)); draw(circle((4.800000,3.000000),3.720215)); draw(circle((4.850000,3.000000),3.690867)); draw(circle((4.900000,3.000000),3.661967)); draw(circle((4.950000,3.000000),3.633524)); draw(circle((5.000000,3.000000),3.605551)); draw(circle((5.050000,3.000000),3.578058)); draw(circle((5.100000,3.000000),3.551056)); draw(circle((5.150000,3.000000),3.524557)); draw(circle((5.200000,3.000000),3.498571)); draw(circle((5.250000,3.000000),3.473111)); draw(circle((5.300000,3.000000),3.448188)); draw(circle((5.350000,3.000000),3.423814)); draw(circle((5.400000,3.000000),3.400000)); draw(circle((5.450000,3.000000),3.376759)); draw(circle((5.500000,3.000000),3.354102)); draw(circle((5.550000,3.000000),3.332041)); draw(circle((5.600000,3.000000),3.310589)); draw(circle((5.650000,3.000000),3.289757)); draw(circle((5.700000,3.000000),3.269557)); draw(circle((5.750000,3.000000),3.250000)); draw(circle((5.800000,3.000000),3.231099)); draw(circle((5.850000,3.000000),3.212865)); draw(circle((5.900000,3.000000),3.195309)); draw(circle((5.950000,3.000000),3.178443)); draw(circle((6.000000,3.000000),3.162278)); draw(circle((6.050000,3.000000),3.146824)); draw(circle((6.100000,3.000000),3.132092)); draw(circle((6.150000,3.000000),3.118092)); draw(circle((6.200000,3.000000),3.104835)); draw(circle((6.250000,3.000000),3.092329)); draw(circle((6.300000,3.000000),3.080584)); draw(circle((6.350000,3.000000),3.069609)); draw(circle((6.400000,3.000000),3.059412)); draw(circle((6.450000,3.000000),3.050000)); draw(circle((6.500000,3.000000),3.041381)); draw(circle((6.550000,3.000000),3.033562)); draw(circle((6.600000,3.000000),3.026549)); draw(circle((6.650000,3.000000),3.020348)); draw(circle((6.700000,3.000000),3.014963)); draw(circle((6.750000,3.000000),3.010399)); draw(circle((6.800000,3.000000),3.006659)); draw(circle((6.850000,3.000000),3.003748)); draw(circle((6.900000,3.000000),3.001666)); draw(circle((6.950000,3.000000),3.000417)); draw(circle((7.000000,3.000000),3.000000)); draw(circle((7.050000,3.000000),3.000417)); draw(circle((7.100000,3.000000),3.001666)); draw(circle((7.150000,3.000000),3.003748)); draw(circle((7.200000,3.000000),3.006659)); draw(circle((7.250000,3.000000),3.010399)); draw(circle((7.300000,3.000000),3.014963)); draw(circle((7.350000,3.000000),3.020348)); draw(circle((7.400000,3.000000),3.026549)); draw(circle((7.450000,3.000000),3.033562)); draw(circle((7.500000,3.000000),3.041381)); draw(circle((7.550000,3.000000),3.050000)); draw(circle((7.600000,3.000000),3.059412)); draw(circle((7.650000,3.000000),3.069609)); draw(circle((7.700000,3.000000),3.080584)); draw(circle((7.750000,3.000000),3.092329)); draw(circle((7.800000,3.000000),3.104835)); draw(circle((7.850000,3.000000),3.118092)); draw(circle((7.900000,3.000000),3.132092)); draw(circle((7.950000,3.000000),3.146824)); draw(circle((8.000000,3.000000),3.162278)); draw(circle((8.050000,3.000000),3.178443)); draw(circle((8.100000,3.000000),3.195309)); draw(circle((8.150000,3.000000),3.212865)); draw(circle((8.200000,3.000000),3.231099)); draw(circle((8.250000,3.000000),3.250000)); draw(circle((8.300000,3.000000),3.269557)); draw(circle((8.350000,3.000000),3.289757)); draw(circle((8.400000,3.000000),3.310589)); draw(circle((8.450000,3.000000),3.332041)); draw(circle((8.500000,3.000000),3.354102)); dot((0,0),red); dot((10,0),red); dot((7,6),red); label((0,0),"$B$",SW,red); label((10,0),"$C$",SE,red); label((7,6),"$A$",N,red); draw((0,0)--(7,6)--(10,0)--cycle,red); [/asy]$

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